As Lambie's answer points out, Jonathon Green's Dictionary of Slang identifies schnip as Yiddish slang, dating to the 1960s.
A Google Books search turns up a couple of earlier instances. From Peter Packer, White Crocus (1947) [combined snippets]:
"How old are you?"
"Getting on for eleven."
"Young as that! Just a kid. I'm twelve. Fancy going out with a shnip like you. That mother of mine and her fancy ideas!"
Packer was born in London but at some point moved to the United States, where he became best known for writing 25 episodes of the TV series Lost in Space.
And from Gerald Kersh, The Thousand Deaths of Mr. Small (1951) [combined snippets]:
"Let go. You're breaking my hand."
"What's the matter? A great big man like you, begging of a schnip like me I should let go his hand? What are we coming to?"
...
Charles Small, sixteen and a half years old, said : “Oh, come on, Father—man to man—tell us.”
If Millie had been present then, I. Small would have thundered: "Man to man? What, so he' s a man already, that schnip? The bleddy cheek of it! Let them learn to wipe their noses before they call themselves men!”
...
"—Quidleright! That was where I met you, Schloi! You was a nothing, then, a bit of a schnip."
Kersh—perhaps best known as the author of Night and the City—was born and raised in England but eventually became a U.S. citizen.
I suspect that shnip/schnip might be more accurately characterized as British Yiddish slang, since the two earliest recorded instances of it that I could find are from English authors, and since I haven't been able to find any mention of it (spelled either schnip or shnip) in any of the Yiddish dictionaries that I consulted. In any case, it has been in use at least since the 1940s, seemingly in the sense of "an insignificant or callow person." Green's definition misses the "excessively young" aspect of the term implied in the Packer instance, the second Kersh instance, and the OP's quote—which comes from Arnold Wesker, The Kitchen (1960). Wesker, by the way, was also from London; he lived in the UK all his life.
A one-word equivalent of shnip/schnip might be whippersnapper, which Merriam-Webster's Eleventh Collegiate Dictionary defines as follows:
whippersnapper n {alter. of snippersnapper} (1700) a diminutive, insignificant, or presumptuous person")
This definition, too, seems to ignore the extent to which the word is associated specifically with presumptuous youth, which perhaps makes it even more apt as a synonym. As for snippersnapper, that word goes back to circa 1590, according to the Eleventh Collegiate. There doesn't seem to be any established connection between snippersnapper and shnip/schnip, however.
Update (August 13, 2020): Is 'schnip'/'shnip' Yiddish?
As a followup to the research I did yesterday, I consulted a couple of online Yiddish dictionaries—Yiddish Dictionary Online and Yiddish Dictionary Lookup (a resource maintained by the University of Kentucky). The former returned no results for schnip or shnip. The latter had nothing for schnip but two entries for shnip:
shnip verb, participle ge...t, cut finely adverbial complement avek, arop
shnips noun, plural in -n, gender m, necktie has diminutive
Neither of these meanings—"cut finely" and "necktie"—has much immediate connection to the slang term schnip/shnip as used in the books cited above. As I mentioned in my original answer, if schnip/shnip in the sense of "insignificant person" qualifies as Yiddish at all, it does so as a slang term of relatively recent vintage, probably localized in the UK. Indeed, it may be a Yiddishized borrowing of the English noun snip, in the following relevant sense (as defined in Merriam-Webster's Eleventh Collegiate Dictionary):
snip n ... 3 : a presumptuous or impertinent person; esp : an impertinent or saucy girl
In the examples I noted earlier, the "[often female] presumptuous or impertinent person" (snip) takes the form of a [male] presumptuous or impertinent nobody or young person (schnip/shnip). But that isn't much of a leap.
Leo Rosten coined the term Yinglish for "Yiddish words, phrases and locutions that are now part of the English language or, because of their incomparable connotations, should be." But Rosten was viewing English (and Yinglish) through a prism of American usage. To characterize a term like schnip/shnip that caught on (at least to some extent) the UK but not (as yet) in the US, a more apt term might be Briddish.
