Congratulations on your new cat.

First you should keep the 2 separate, especially when you can't monitor their interactions.

If the new male is mellow, I have no doubts that he will win the resident female over. Females are a little more stubborn about accepting new-comers than males are.

As for a time frame, please don't think all cats will get friendly in the same amount of time. I have seen instant like......I have seen months pass before the cats co-exist. But I'd say your average would be 1 to 3 weeks.

We'd love to see and hear more about your kitties.