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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.
An idea: you should be able to construct a NAND gate, so you can then build a XOR gate. With XOR and AND you can build a half-adder. Combine half-adders to build a full-adder. That would be a start at least.
NAND and NOR are basic building blocks for other gates so chances are Turing completeness is just around the corner.