You need NOT and one of AND or OR to be able to do all binary logic.
This is DeMorgan’s Law, basically.
However, this is not sufficient for Turing completeness.
For that you also need random (or reducably equivalent) access
(theoretically) infinite memory.
Odds are, you’ll be able to build a flip flop (a D flip flop
is built using NANDs, so it’s straightforward) using
the available logic gates. From those, you can build a
register, and with enough of those you’ll be equipped
to build some simple programs.