Here's my experience.
When I had a puzzle accepted by the NYT, I was unaware the wait to see it published would be so long (about two years for me). But it stands to reason since Will receives so many. There was no correspondence between us during that time (no edit requests for example), and I was starting to worry if maybe it had gotten lost in the shuffle, or so far down the list that it would be a very long time. But at last I got the email a few months ago that he was going to run it. He will let you know the week before that it's going to be published. You can expect a check a few weeks later, depending on your location.
As for the LAT, I have a lot more experience. There's apparently no forms to fill out, though i would assume rights are tacitly signed over upon acceptance. Sometimes Rich will ask for a revision, and on average I would say it takes about a month for him to get back to you after you send it back (he has a long list of manuscripts himself!). The LAT's wait time for publication, for me, is more in terms of months than years, sometimes 1 or 2, other times 4 or 5, possibly to avoid close repetition of theme entries, or because he just has a lot of other crosswords in ahead of yours. Either way he'll let you know a day or two before the particular month of publication begins. As with the NYT, the check will arrive a few weeks after the puzzle appears.
So far, I've never had a crossword accepted that was never published.
I hope I've answered all your questions. If I think of any other pertinent information, i'll let you know