What would happen if I had a type rating for an Airbus A320 and applied for an airline that has an A320 and other aircraft?
No. You'd take the same initial course as everybody else, learning that airline's procedures &amp; checklists for that aircraft.  Having a type rating shows the hiring people that you're able to pass the training course &amp; checkride for that level of aircraft, and that's good to have. But beyond making the aircraft systems part of the syllabus easier for you, having the type won't change your training.

It also probably won't affect which aircaft you initially get. Those are based on where the airline needs pilots, and then (typically) on some sort of bid process. The airline presumably expects all its new pilots to be able to complete whichever training course they get, so your type and/or prior experience is unlikely to drive your initial assignment.

For a new, small airline in need of hiring captains off the street, things might well be different.