According to Daniel Defoe, The Political History of the Devil (1726), the attribution of oldness to the Devil reflects his relative experience and wiliness:
Well might the Text call it listening to the Council of the young Heads ; that it was indeed with a Vengeance ! but those young Heads too were acted by an old Devil, who for his Craft is called, as I have observ'd, the Old Serpent.
Defoe says that the Devil is first called "the old Serpent" in Revelations xii: 9. The Genevan New Testament (1557) gives this wording for that passage of the Bible:
And the great dragon, that old serpent called the deuyl and Satan was cast out, which deceaueth all the world. and he was cast into the earth, and his Angels were cast out with him.
It is this name for the Devil that appears earliest in the company of "old." For example, from Bishop Poynet, The Real Reformation Catechism of 1553 (1553):
Scholar.—In the serpent's head lieth all his venom, and the whole pith of his life and force. Therefore do I take the serpent's head to betoken the whole power and kingdom, or more truly the tyranny, of the old serpent the devil. The Seed (s Saint Paul doth plainly teach) is Jesus Christ, the Son of God, very God an very man : conceived of the holy Ghost : engendered of th womb and substance of Mary, the blessed pure and undefiled maid : and was so born and fostered by her as other babes be, saving that he was most far from all infection of sin.
Master.—All these foundations that thou hast laid are most true. Now therefore let us go forward to those his doings, wherein lieth our salvation and conquest against that old serpent.
Likewise, William Crashaw, The Parable of Poyson. In Five Sermons of Spirituall Poyson (1618) refers to "the old Serpent" twice:
And as Christ saith of miraculous Faith, that by the vertue of it, they shall take up Serpents and if they drinke deadly poison, it shall not hurt them : So is it true of iustifying Faith, that whosoeuer beleeues, shall wrastle with Sathan the old Serpent, and be safe, and the drink of the deadly poison, which his sinnes gaue him in the time of ignorance and impenitencie, shall not hurt him : This was prefigured in the brasen Serpent, which as it deliuered the people from the biting and stinging of fiery Serpents ; so our Sauiour lift up upon the Crosse, saues all them that looke up to him with the eyes of Faith, from all the fiery and poisoned darts of the Diuell, whether darts of temptations here or of damnation hereafter.
Wee here learne how naturall and proper it is for wicked men, to mock, abuse, and persecute Gods children : For they being Serpents, euen the spirituall generation of the old serpent Sathan, therefore by the sentence of God himselfe, they are at an euerlasting odds & enmity with Gods people.
And from George Downame, A Treatise of Justification (1633):
And as Moses lifted up the serpent in the wildernesse, that they who were bitten of the fiery serpents by looking on the brasen Serpent, which was a figure of Christ, should be healed : even so the Sonne of man was to be lifted up upon the Crosse, that whosoever being stung by the old serpent the Devill, looketh upon him with the eie of a true faith, that is beleeueth in him, should not perish, but have eternall life, which truth is acknowledged by the Master of the sentences : [Latin quotations omitted].
And from Henry Ainsworth, Annotations upon the Five Bookes of Moses, the Booke of the Psalmes, and the Song of Songs, or Canticles (1639):
And as no salve or medicine could heale the bodies of those that were bitten: so can no worke of man cure the biting of that old Serpent or sting of sinne, but the venom thereof rageth and reigneth, tormenting the conscience unto death.
Many instances of "the old Serpent" as a nickname for the Devil appear in later 17th-century writings
"Old Nick" appears to be the oldest of the three main terms that the OP mentions. From Edmund Gayton, Pleasant Notes Upon Don Quixote (1654):
But that which sticks I'th' stomack of our Don,
Like a good Meale (if ever he get one)
Was the redeemed Slaves ingratitude,
Whom he enlrg'd and gave full latitude
Of Leg and Arme, which they uncivill Devils
Employ against the Rescuer from their evils.
Whom thinke you in this Fact was Paramount,
But that unlucky Rogue Gines Passamont ?
Whom though the Gallies misse, yet for this trick
lle warrant him a Passe-port to old Nick.
And from "A Song" in Rump: Or an Exact Collection of the Choycest Poems and Songs, Relating to the Late Times (1662):
The Swearer there shall punish be still,
But Drunkennesse private be counted no ill,
Yet both kinds of lying as much as you will,
For Round-heads Old Nick stand up now.
Francis Grose, A Classical Dictionary of the Vulgar Tongue (1785) reports a folk explanation for the terms "Old Nick" and "Old Scratch":
OLD NICK, the devil. The following ludicrous reason is given for this appellation, and that of old scratch : the angel first employed in forming women, had forgot to cut their parts of generation, which the devil undertook to do by the following contrivance, he placed himself in a kind of sawpit, with a scythe fixed to a stick, in his hand, and directed the women to straddle over it ; the pit being too deep for the length of his instrument, he gave the tall women only a moderate scratch, but the little women by the shortness of their legs coming more within his reach, he maliciously gave them monstrous gashes, or nicks, whence he was called old scratch, and old nick.
That same source identifies the terms "Old Harry," "Old One," and "Old Roger" as additional then-current nicknames for the Devil.
One of the earliest occurrences of "Old Scratch" in a Google Books search is from "The Story of Jack Spriggins and the Enchanted Bean" in Christmas Entertainments (1740):
You have a thousand Jack Catches now attending you without with Halters and Hatchets to make an end of her [Jack's grandmother's black cat], when your Honour pleases to direct her Execution ; or else you have a fiery Dragon gaping for her, if you give but once the Signal for her Death : This Box, great Sir, bears you the absolute Power over her, over us, over Old Scratch or Nicholas the Antient.
One year later, Henry Fielding, An Apology for the Life of Mrs. Shamela Andrews (1741):
You see, Madam, says she [Mrs. Jewkes], I carry the Marks of your Passion about me ; but I have received Order from my Master to be civil to you, and I must obey him : For he is the best Man in the World, notwithstanding your Treatment of him. My treatment of him; Madam, says I ? Yes, says she, your Insensibility to the Honour he intends you, of making you his Mistress. I would have you to know, Madam, that I would not be Mistress to the greatest King, no nor Lord in the Universe. I value my Vartue more than I do any thing my Master can give me ; and so we talked about my Vartue ; and I was afraid at first, she had heard something about the Bantling, but I find she hath not ; tho' she is as jealous, and suspicious, as old Scratch.
John Bartlett, Dictionary of Americanisms (1848) notes that in informal American speech the word old had by the late 1820s acquired the sense of wiliness:
OLD. Crafty; cunning. Used in vulgar language.—Webster . When a person attempts to get the advantage of another, and is frustrated in the attempt by the sagacity or shrewdness of the other, the latter will say, 'I'm a little too old for you,' meaning that he is too cunning to be deceived by him.
This sense of old may or may not have come from the longstanding association of old with the Devil; a more neutral reading of the phrase "I'm a little too old for you" is that it means "I'm not as naïve as you think."
To sum up, the association of the Devil with old may in the first instance be connected to the biblical reference to "that old serpent called the devil and Satan," and subsequently (and more generally) to the Devil's ancientness and cunning.