I do not know if I have a cold start timer on it. Would you happen to have a picture of it on the engine? Or the general area? Then I will look tomorrow morning before I go to school. It's just a rebuilt 22RE.
You will need to test the cold start injector, as well as the timer, to make sure you don't have a wiring issue.
Cold start timer switch(black connector) is located at the front of the engine, below the intake, next to the water temp sensor(green connector).
Pic is from an '86, but should be in the same place.
To test