within a couple of days. If she is a virgin queen, it will take a week or two for her to get mated and start laying eggs. If you find eggs, you know you have a queen. Some beekeepers will catch the queen to ensure she doesn't fly off again, but there are other ways to deal with that if you think they might (can happen with one of your own swarms that you capture a few feet away - they don't want to go back to the same place necessarily.

If there is a queen problem thereafter, it is easily diagnosed and you either combine the colony with another one, buy a queen, or help them to make their own (by putting in a frame of brand new eggs. I have done all three if you have questions about those options (are you a beekeeper - you ask about splits?).

We haven't really done much with splits. The weather here makes it a bit tricky (warm weather going to really cold weather in Mar/Apr). We should have this year because swarming was so bad (splits reduce swarming).