I’m not exactly sure of your question. As Aart said, you do not need a rooster for a hen to lay an egg. They will lay eggs whether a rooster is around or not. But you do need a rooster to have fertile eggs.
If your question is when are the eggs fertile, it takes about 25 hours for an egg to go through the hen’s internal egg-making factory. It can only be fertilized in the first few minutes of that journey. That means if a successful mating takes place on Monday, Monday’s egg cannot be fertile. Tuesday’s might but I would not count on it. Wednesday’s egg should be fertile.
A rooster does not necessarily mate with every hen in his flock every day, but the hen can store his sperm for two weeks or maybe even three weeks in a container near the start of her egg-making factory. So a rooster just needs to mate a hen once every two weeks for her to produce fertile eggs.
You do not need a rooster around for a hen to go broody. A lot of hens will never go broody and you cannot control when they do, but having a rooster around will not have any effect on that. Either they go broody or they do not.
If I misunderstood your question please try asking again.