Dumbledore strikes me as the type who would keep the prophecy to himself, particularly the most telling bits of it. He might stretch the truth and tell the Potters or Longbottoms he had inside information that they were being targeted, but I just don't know if he would just outright tell them and worry them needlessly. He didn't tell Harry, after all. Plus, knowing a prophecy affects the outcome.
That said, I think an author can do what they want with this scenario as long as they write it well and believable. You could totally write a story in which Dumbledore tells them, or a story in which he doesn't. It just has to be fleshed out, explained, and believable. It might be neat to have him tell James but not Lily.
As for the timeline, I again think there is room to fudge it. I personally don't think the prophecy was made before either Alice Longbottom or Lily Potter became pregnant, so it would have to be after October/November, 1980. It was a 'cold, wet night' when Dumbledore was interviewing Trelawney so it could be winter or it could be spring. I wrote a story where it was spring myself.
I'll have to think more on how Snape fits in. The HPL has him warning Dumbledore in summer 1980 but then starting at Hogwarts in 1981. It's rather confusing since one page says in the fall and one page says after the Potters are killed. >.<
This only leaves us with the secret-keeper, problem. Perhaps Pettigrew knew for an entire year before he finally cracked and changed sides.
Sirius Black does accuse Pettigrew of passing information to Voldemort for over a year. However, the Fidelius Charm was not cast until a week or so before Halloween (Oct. 24 according to the HPL) so it only took a week for Peter to give up the Potters. Perhaps he was tormented.
Hope that helps - since there is no solid canon, you have some options! Good luck!