Turing-completeness, Conway's Game of Life and Logical Gates
A form of conditional repetition or conditional jump (while, for, if and goto)
A way to read and write to some storage mechanism


The computer you typed this message onto is Turing Complete (well, sort of), yet the only tool it uses is logic gates.  I can give you a simple(ish) example of each:

For the first requirement, processors can use multiplexers to create conditional jump options.  Multiplexers are basically just a bunch of AND gates.

Number 2 is a bit more complex.  Computer memory consists of logic gates that feed back into themselves.  If you study memory, you begin by studying latches and flip-flops.  Here is a simple latch that stores which of the two input lines on the left turned on most recently:



Thus, it stands to reason that, if you can imitate logical gates, you can create a Turing-Complete machine just as you would if you could simulate a Turing Machine.  So, it may be that you don't need to show how to simulate a TM in GoL nodes, but instead take a step to show that logical gates are sufficient to be Turing Complete.

I have one further thought for you that might make your life simpler.  You don't need every gate.  NAND and NOR are each sufficient, on their own, to create every other logic gate.  You can easily do a search like "nand logical completeness" to find the logic gate diagrams to show this.