The best answer is simply that "it depends.."
It depends on which bank it is; how much presence they have in the area; if they already have relationships with brokers or need to establish them; if they are a big bank or a small one; if the loan was government insured (FHA/VA, etc) or backed by Fannie Mae, etc, etc. You get the idea....
So - the most reasonable expectation should be something like 1 to 3 months after sale, but it can vary - a lot.
Please let me know if you'd like help looking into all this in more depth.