I've done a little bit of research about this "shtook" slung term and it looks like the Yiddish origin cannot be ruled out so easily.
To sum it up, I've come to the conclusion that the most probable origin is an abbreviation of the phrase shtuck dreck which means piece of crap and it comes from the Jewish community.
Here is some supporting evidence.
1. Wikipedia has an list of article titled "English words used by English-speaking Jews. One entry reads:
shtick dreck – literally "a piece of dirt" (see Dreck), but usually
applied to a person who is hated
because of the antisocial things he
has done: "He's a real shtuck dreck."
Possibly shtick dreck: a piece of
crap. Cf. German Stück Dreck.
NB: In German: Ein Stück means "a piece of" and Dreck in Yiddish means "excrement".
Hence the often seen deep in "in deep shtuck". Speaks for itself.
2. The slang dictionary also postulates a Yiddish origin.
shtuk/shtook/stook/schtuk in trouble. A very widespread expression which moved from a
restricted demi-monde and theatrical
usage to common currency in the
mid-1960s, partly through its use in
the entertainment media. Shtuk in its
various spellings is Yiddish for
difficulties. ‘In shtuk’ often refers
to financial difficulties.
3. I also found in the usenet archives at Google (formerly dejanews) the expression used in the soc.culture.jewish news group:
They really do stick to their shtuck drek.
4. One possible reason why the word might be deemed as non Yiddish is because it is supposed to come from London (as opposed to NY). However I also found this interpretation:
...are ubiquitous in the East End (London)
but seem to have been particularly strong around
the Bethnal Green/Spittalfields market areas.
There is (or was) a strong Jewish immigrant
presence in that quarter. It seems quite
possible to me that 'shtook' is a joint
So despite the Partridge Slang dictionary I'd go for the Yiddish/German origin.