Pushing truck to start is not for a condition of "no turn over". You need to take the push button off starter and examine or replace. Or take off starter all together and bench test. So far seems like dead push button or dead starter IF IF IF you are sure battery is good and cables and clamps are clean. Let us know if lights and such come on. Be sure motor is grounded to battery Neg.

Starter is grounded thru it's mounting bolts to motor.

You may want to push the button with a stick by hand to be sure it is being pushed far enough down.
The transmission has nothing to do with this. But I'm not sure if you are saying you have a transmission issue or not.

This is probably going to be an easy fix provided you have rocked the truck to see that motor is not frozen. Check oil for milky look.

All suggestions based on:
1. no click
2. no noise
3. no nothing
4. good battery that has been tested for voltage/shorts. (new? as in new or another?)
5. good clean cables and clamps
6. starter is grounded to battery thru normal method
7. button on starter pushed by hand

Right is right, even if everyone is against it, and wrong is wrong, even if everyone is for it. - William Penn