Cat will not cause a no start. Its possible they broke it when they pulled the trans. In the auto repair world, trans installers are at the bottom of the food chain, some of these guys are real animals and places like AAMCO tend to get the worst of the lot.
Keep in mind, they probably dont have a single actual techinician on the payroll, they have a bunch of monkeys out front riping transmissions from cars, and some guys in the back rebuilding them at light speed. They have absolutley no clue how to diagnose an engine problem, but they will all fancy themselves mechanics and will be happy to tell you how to get the most from your 1968 Nova.
They broke it, they fix it. The car ran when you droped it off, it needs to run when you get it back. If they have to pay another shop to make it run, then thats what they have to do. I have had really horrible luck with AAMCO and I have had to repair some incredible butcher jobs they have done on trans installs.
Don't let them replace the exhaust, if you really do need a cat, you are better off at Midas than letting AAMCO replace it. Do not get nasty with them, just be firm abut wanting your car back running properly. They do have your car as a hostage afterall.
Most likely they forgot to plug something in.