What happens if uninsured motorist coverage was used by my car insurance company & later the hit and runner got caught?

First. It isn't "uninsured motorist" coverage that covers your vehicle, it would be comprehensive....................But..............You don't have to be concerned about that anyway. Your insurance company will go after him and they will likely not notify you about whether they were able to collect or not.

the insurance co will take care of getting their money back. Leave it to them that is what you pay for.

Uninsured motorist cover has nothing to do with any hit and run.

Your insurance company will go after the insurance company for the money they paid you. That's called a subrrogation claim. They will also try to get you back the deductible you had to pay for the vehicle damages under the Collision coverage

Your Insurance Company would subrogate or collect from the liable party.

your insurance company can ask for compensation from him. It's always good when they do

The insurance company sues the hit & runner. You have no obligation to pay back any money.

That happens all the time, and yes they will go after the owner of the offending vehicle.

The process is called subrogation, and every insurance company has a whole department dedicated to it. They go after the guilty party in court, and they do it relentlessly and ruthlessly. When they recover the money, you're compensated first (any deductible you paid) and they get the rest.

Yes, your insurance company should try to collect the cost of your repairs from the other driver on your behalf.