Can the M1 Abrams tanks work without computers?

Every US military combat vehicle is capable of operating in manual mode. It is just much slower. As long as the vehicle is still running and the batteries still have power, any model of the M1 can still fight. There are manual back up sights. Manual firing procedures. Manual loading procedures (which is basically all manual as it is).

I don't know the answer to your question for certain, but I can give you an educated guess. Modern military equipment is usually required to have some sort backup system that can be used if the primary system fails. For a device as expensive and complicated at an Abrams tank they probably have backups for the backups.

I'd suspect it can still fire a round, but without computer assisted aiming wouldn't be able to hit a moving target while it's moving. You would need to aim/fire the old fashion way using a site. With a tank unless firing point blank would mean the tank needs to be stopped to aim/fire.

Your writing a story, you can make up whatever backup technology you want. If it was me writing the story the tank could still be driven and shoot rounds, but it would function like a 1940s Sherman - all manual w/hydraulic assist.

My question is, 20 years after the zombie apocalypse or whatever, when your premise is that computers can't work, how is a tank with a gas turbine engine operating AT ALL?

Yes. it can.

aka, GAS, the master blaster, and the manual cranks.