Virtually all home insurance policies today cover plumbing leaks that are sudden and accidental. In other words, you're covered for a pipe that abruptly springs a leak, as opposed to one than trickles out water over months or years.
Older policies also cover that long-term type of pipe failure, provided that you could not know about the problem until you discovered it.
So, the answer to your question hinges on the type of policy and type of leak you have. There is no uniformity, though even a policy that covers leaks of any kind will significantly limit or exclude any related mold damage. Your agent should be able to explain your coverage to you.
Let's assume you do have good leak coverage. In that case, your policy will pay for: the cost to tear out the wall to get at the leak; all resulting water damage, including mold repair that's covered; and the expense of repairing the wall and replacing tiles, etc. In short, the only part of a covered loss that's not covered is the cost of repairing the leak itself.
As for your question of whether you can still file a claim after the repair has been fixed, that depends on how long it has been since the work was done. Technically, you have violated a condition that gives the insurance company the right to inspect damage before repairs are made. However, in practice, most home insurance companies will not deny your claim solely based on late reporting. But when insurers can't inspect the damage themselves, it can be tough sledding for the policyholder.