Early mentions of ‘nannicock’
From Thomas Lewis Owen Davies, A Supplementary English Glossary (1881):
NANNICOCK, a silly, affected person. See H. s. v. nanny hen.
Hee that doth wonder at a weathercocke,/And plaies with euery feather in the wiude,/And is in loue with euery nannicocke. Breton, Pasquil’s Fooles-cappe, p. 23.
Gordon Williams, A Dictionary of Sexual Language and Imagery in Shakespearean and Stuart Literature (2001):
In [Richard] Brome, New Academy (1623–40) V.i, a wife is called 'your Nansie Nanny Cock', allowing the husband to slip into unconscious bawdry when declaring that formerly he would not have denied her 'any thing, if my Cock had but stood upon't'.
The actual quotations from New Academy involving “Nanny Cock” are as follows. From Act II, scene 1:
Hannah [“Camelions wife”]. You shew your love, Rafe.
Rafe Camelion [“an uxorious Citizen”]. So I hope I do Nan./My cock, my pity nittle nansie cocksie,/Do I not shew my love when I deny thee/Vnreasonable requests?
Camelion. Hon soit qui maly pense. My Cock, my Nansie Cock, my Cocksie Nansie, Kisse me, and use thine own conscience:
From Act 3, scene 1:
Hannah. But if you will not stay to reade this Letter./You shall not deny me one thing.
Camelion. What is it, quickly? my sweet Nanny Cock.
From Act 5, scene 2:
Camelion. I dare not look on her, lest I be tempted/To yield unto my shame and my undoing.
Valentine Askal [“Hannah’s half-brother”]. Will you not heare your Cock, your Nansie, Nanny Cock.
Hannah. Time was you would not ha’ denied me that.
Camelion. Nor any thing, if my Cock had but stood upon’t./Such was my love, but now,
Hannah. But now y’are jealous.
Valentine. Do'st think I did not know thee.
Hannah. No sir, nor would I that you should,/Till l had foil’d you in your course,/And had my will to make my huſband jealous.
Camelion. My Cock, my Cock again, my Nanny cock,/Cock-all, my Cock-a-hoop, I am overjoy’d,/See, see thy father too.
And from Catalogue of the Pepys Library at Magdalene College, Cambridge, volumes 1-2, we have this entry from the 1600s:
Title: THE/Frightned People of Clerkenwel,/Being an Account of how a COW Ran into the Church at Clerkenwwel in Sermon time,...
Tune: To the Tune of, In Rome there is a most fearful Rout.
1st lines: IN Clerkenwell-Church there was a Rout/Last Sunday the People like Bedlams run out,
Refrain: But Nannicock my poor Cow, poor Cow;/But Nannicock my poor Cow. (with variations)
Date: 1680 (Simpson); Commonwealth (1649–1660); Restoration (1660)
‘Nanny cock-a-thaw’ remembered in the mid-1800s
From Joseph Wright, English Dialect Dictionary, volume 4, M–Q (1903):
NANNY, sb. Var. dial. Uses in Sc[otland] and Eng[land] ... 1. In comb. ... (3) –cock-a-thaw, a spark ; see below ; ... [notes:] (3) e[ast] Lan[cashire] When playing at forfeits, it was usual to hold either a piece of paper or wood which had been burnt, and on which a spark still remained. Before the spark went out, we were obliged to repeat the following: ‘Nanny Cock-a-Thaw, Nine sticks, nine stones Shall be o’ thy bones If thou let Nanny Cock-a-Thaw faw [fall].’ Of course, if the spark went out before the lines were repeated, a forfeit was demanded, N[otes] & Q[ueries] (1866) 3rd S[eries] ix, 87.
The relevant Notes & Queries source for this usage is here. It identifies the source, Gibson, as living “In East Lancashire, in the ancient borough of Clitheroe and neighbourhood.”
A very similar account appears in Thomas Bushfield, “Memorials of Sixty Years Ago—at Ashford in the Water [Derbyshire],” in The Reliquary and Illustrated Archaeologist (July 1867):
Nothing is more delightful and cheering in after life than the supreme pleasure of recalling the saying and circumstances of our early days. ... Now for the evidence of from my own memory!—my very earliest remembrances. Not one of them have I ever read, however familiar they may seem to your readers. No doubt many of them are known almost universally, such as “Pat a cake, pat a cake, baker’s man;” “Bye baby bunting;” “Cuckoo! Cherry tree!” “The cuckoo is a pratty bird;” “Baa lamb! Black sheep!” “To bed, to bed, says Sleepy Head;” “Little Tommy Linn;” “Robin a Bobbin;” and others.
Little Nanny Cock a thaw;
Suppose if I should let her fa’?
Nine sticks, nine stones,
Shall be laid upon thy bones,
If tha’ lets Nanny Cock a thaw fa’
This is said on the occasion of a winter’s evening’s amusement among the young, arising from the following performance—a lighted stick is placed in the hand of one of the party, while so held the above words are said, the lighted stick is then passed on to the next, and so on; the one who happens to be the holder when the fire dies out pays a forfeit.
Instances of ‘nanny-hen’ through the years
Davies's Supplementary English Glossary (the first work cited in this answer) suggested a connection (never specified) between nannicock and nanny hen. I decided to look into the history of nanny hen, and found quite a bit of coverage of the term.
From Halliwell, Dictionary of Archaic and Provincial Words, second edition, J–Z (1852):
NANNY-HEN. As nice as a nanny hen, i. e. very affected or delicate. Cotgrave has the phrase, “as nice as a nunnes henne.”
Women, women, love of women
Make bare purs of some men.
Some be nyse as a nanne hene,
Yit al thei be nat so;
Some be lewde, some all be shreude,
Go schrewes where thei goo.
MS. Lambeth 306, f. 135.
In a note attached to the Lambeth poem, the editor of an edition of Adam of Cobsam’s The Wright’s Chaste Wife (ca. 1462) from the 1860s offers this commentary:
The Rev. J. R. Lumby first told me of the proverb ‘As white as a nun’s hen,’ the nuns being famous, no doubt, for delicate poultry. John Heywood has in his proverbs, 1562 (first printed, 1546), p. 43 of the Spencer Society's reprint, 1867,
She tooke thenterteinment of the yong men
All in daliaunce, as nice as a Nun's hen.
The proverb is quoted by Wilson in his Arte of Rhetorique, 1553 (Hazlitt's Proverbs, p. 69).
John Farmer, The Proverbs, Epigrams, and Miscellanies of John Heywood  (1906) quotes the dalliance couplet from Heywood and then offers this note on the phrase “nun’s hen”:
HEN, (a) “as nice as a nun’s hen” (52c), a very ancient proverbial simile : ? nun = (a) a variety of pigeon having its head almost covered with a veil of feathers; (b) the smew; or (c) the blue titmouse — most likely the last. “Women, women, love of women, Make bare purs of some men. Some be nyse as a nonne hene, Yet al thei be nat so; Some be lewde, some all be shreude, Go schrewes where thei goo.”—Satirical Verses on Women (1462). “I have the taught dyvysyon between Frende of effect, and frende of countenaunce; The nedeth not the gall of none hen That cureth eyen.”—Lydgate, Proverbes (c. 1520). “I knewe a priest that was as nice as a Nonnes Henne.”—Wilson, Art of Rhetorique (1562).
The reference in Thomas Wilson, The Art of Rhetorique (1553) is quoted at greater length in Wilhelm Horn, “Lichtenberg über nasty” (March 1909), in Anglia: Beiblatt, volumes 19–20 (1909):
An other speakes so finely, as though he were brought vp in a Ladies Chamber. As I knew a Priest that was a nice as a Nonnes Henne, when he would neuer saie Dominus vobiscum [“the Lord be with you”], but Dominus vobicum [“Lord yourselves”]. In like manner as some wil now say, the Commendementes of God, blacke vellet, for Commaundementes and black veluet.
A modernized version of Wilson, using the spelling “nun’s hen,” appears in Peter Medine, ed., Art of Rhetoric (1560) Thomas Wilson.
From a note by Walter Skeat in Notes & Queries (1866):
As Nice as a Nun's Hen" (3? S.jt. 169.) The word fastidious very nearly expresses the sense of nice here. The priest alluded to was fastidious and mincing in his talk ; and, by a sort of pun, was said to be as fastidious and particular as a nun's hen ; according to a proverb in the north, which makes a nun’s hen to be something peculiarly delicate and pure. The following quotation well exemplifies this:— the author then supplies the “Women, women, loue of women” verses again, but with the third line given as “Some be nyse as a nonne hene,” instead of “a nanne hen.”
From a 1963 edition of Heywood, A Dialogue of Proverbs, the editor Rudolph Habenicht offers this note [combined snippets]:
as a nyce as a nuns hen. Literally as fastidiously chaste as a nun’s hen (here used ironically). However, this proverbial phrase is ambiguous in meaning, for nyce may mean either fastidious or “wanton” (OED); and a nuns hen is possibly the hen of a nun, or, following Tilley, a blue titmouse, "whose crest of white feathers gives it a nunlike appearance” (N 353). In either event, it does not necessarily mean “modest.”
From the “Proverbial Similes” chapter of John Ray, A Collection of English Proverbs (1678):
As nice as a nuns hen.
The wording is common enough even in 1852 to appear as one of the similes that a character who speaks almost entirely in clichés uses in William Mountford, Thorpe, a Quiet English Town, and Human Life Therein (1852):
“Yes, sir. He and Mr. John, — they are hand and glove. He says now he gets to be as hungry as a church mouse. Though I know once at his food he was as nice as a nun's hen. But he knows now that hunger is the best sauce, and that medicines are not meant to live on. And so he will soon find that health is better than wealth.”
And from “Miscellany,” in New Eclectic Magazine (August 1869):
These, and a thousand other scraps of witty wisdom, the homely traditions of the past, and are intelligible enough. But who will explain some of our adages? Why should a man be “ as deep as Garrick,” or a prim young lady “ as nice as a nun's hen”? What is the special wisdom of the person who “knows how many beans make five”?
From Thomas Wright, Dictionary of Obsolete and Provincial English (1857):
NANNY-HEN. As nice as a nanny-hen, affectedly delicate.
From John Farmer & William Henley, Dictionary of Slang and Colloquial English (1905):
Nanny-hen. As nice as a Nanny-hen, very affected, delicate.
From Notes & Queries (1917) [combined snippets]:
"As nice as a nanny-hen" is a phrase given by the 'Slang Dictionary.' What is a "nanny-hen"? The word does not seem to be known to any other dictionary.
(iv.) In some cases I have reason to suspect that the form given by collectors is a spurious one. “As plain as a juggem ear” in Hazlitt is a case in point. The word "juggem" puzzled me very long, until I discovered that the phrase must be quoted from Bohn's ' Handbook of Proverbs,' p. 320 (ed. 1855), where we read: " As plum as a juggem ear, i.e., a quagmire, 58.” When transcribing this for his book, Hazlitt must have had his eye on the preceding line: “As plain as the nose on a man’s face. Hence the misquotation. But the matter does not end there. At p. 57 of Bohn's ' Handbook' (58 must be a misprint) there is: " As plum as a jugglem ear." This is also found in Ray, 1768, and probably also ibid., 1678, and no doubt is a misprint for "juggle-mear " or " juggle mear," which is the form given by Lean, who nevertheless also copies Bohn. In the same way, “nanny-hen” may be the result of misinterpretations misquotations or misprints. As the works of reference at my disposal in most cases give little or no information, I must appeal to the readers of 'N. & Q.' for their kind help. I shall be very much obliged for information concerning the correct form, the meaning and application, the origin, and the frequency in modern English, of intensifying similes.
From Jane Mills, Womanwords: A Dictionary of Words About Women (1992) [combined snippets]:
In the C16th and C17th the phrase nice as a nanne's, nanny's or nun's hen meant very affected, delicate, prim or fastidious (see PRUDE). In the C17th nanny became a low colloquialism for a whore (who was, presumably, considered to be none of these things). In the same century a nanny-house or nanny-shop was slang for a brothel. It has been conjectured that in this sense nanny derives from the female personal name Nan, a pet form of Ann(e) which was popular among the British serving class at a time when female servants or MAIDS were often regarded as the personal sexual property of their male employers. For a brief period in the early C18th, nan was a synonym for a serving maid. Another explanation for the sexual pejoration of the word is that, like NICE - which also meant both fastidious and WANTON in the C17th - a whore might affect or adopt the behaviour of a delicate, prim woman. It was yet further ‘proof’ of female wile and deceit.
We have a number of references to variants of nanny, nanne, nonne, or nun’s hen over almost 500 years, including quite a few citations of “nice as a nun’s hen” as a folk simile in common use in England during the 1800s. Also common in parts of England in the 1800s is the phrase “nanny cock-a-thaw,” as part of a fireside incantation in a children’s play rhyme. But otherwise, the only examples of nannicock seem to be the instance in Breton’s “Pasquils Fooles-Cap” from about 1600, the Kirkenwell cow song from the middle to late 1600s, and the odd occurrences of “Nanny cock” as a term of endearment in Richard Brome’s New Academy, by 1640.
The term nanny-hen seems much more common historically than nannicock, but I have not seen any dictionary that directly argues that one arose out of the other or as a gender-specific extension of the figurative meaning of the other. Perhaps the most interesting suggestive information I found involves two words mentioned in Joseph Wright, English Dialect Dictionary, volume 4, M–Q (1903), which I cited earlier in connection with “nanny cock-a-thaw”:
NANNICK, v. and sb. Glo[ucester] e[ast] An[glia]. Also written nanick Suf[folk]; nannack e[ast] An[glia]; nanack Suf[folk]; nannock [e[ast] Suf[folk]; nannuck Suf[folk]; and in forms nonek e[ast] An[glia]; nonnak Suf[folk]; nonneck nonnock e[ast] An[glia]; nonnuck e[ast] An[glia] ... 1. v. To play the fool; to play when one should be working; to idle away one’s time; to fidget. [Examples omitted.] Hence Nannicking, ppl. adj. full of apish tricks; trifling. [Example omitted.] 2. To do light, irregular work; to change one’s employment frequently. [Examples omitted.] Hence Nannucken, ppl. adj. changing about. [Example omitted.] 3. sb. An idle whim; a childish fancy or fear: a jibe. [Examples omitted.] 4. Light, irregular work. Suf[folk]. 5. A valueless trifle, a knickknack. [Citations omitted.] 6. A bungler. [Citations omitted.]
NANNIKIN, ppl. adj. Ess[ex] In comb. Nannikin-job, a piece of work requiring neatness and delicacy. [Citation omitted.] Cf. nannick.
The most interesting thing here is the echo of the nanny-hen’s fastidiousness and affected delicacy in the nannikin-job’s requirement of neatness and delicacy. Still, none of the sources I found draw any connection between nannikin (or nannick) and nanny-hen—or for that matter, with nannicock.