All lights work but no turn signals at all, fuses are good.

Did they splice into your signal lights when they installed your autostart? I don't think they should have spliced into your signals for that unless they were installing an alarm as well.

If you have an alarm, does it flash your lights when you arm your car? When you use the signal lever, does it work for either your left or right signal? Do your hazards work?

Your electronic flasher module may have shorted (it'll be located under the dash near the steering column and will be labeled)