I had an idea where James and Lily weren't killed when Voldemort came to attack (have James be banished through a wall in the house which Voldemort subsequently ignored and Lily petrified as malicious compliance towards Snape in a "she's technically not dead"), instead when the Potters finally recovered and found that Voldemort was defeated by Harry they knew that something was up.
So as the story develops, it's revealed that they left it up to Magic as to what price they must pay for having Harry survive. They knew a life for a life was the usual cost and either would happily pay, but with both of them alive and healthy, Harry alive, and Voldemort gone; things didn't add up. They are forced to ship Harry to the Dursley's and pretend that their son was killed in Voldemort's attack to provide protection from the possibly still alive Voldemort and put the fact that Lily had remaining family under a Fidelus which James held. This double blind means that no one is even searching for Harry and that the fame of the event goes to the Potters. Immediately after they recovered, Lily and James pointed out Peter as the betrayer, recovered Sirius from Azkaban and assigned him to be the go between them and the magical world and Remus the go between them and the muggle world while they hide in the muggle world apart from Harry.
Their focus is trying to find out what went wrong with the pact they made with Magic. They eventually learn that it took the fertility from them both instead of their lives since Magic couldn't permanently kill Voldemort. They are then focused on discovering by which mechanism Voldemort has survived but don't have the spread and connections that Dumbledore so it's much slower for them.
We see a less abused and neglected Harry who has a better outlook on life since the Dursley's couldn't do that they did to canon Harry; this means that we have a more academically focused Harry with much better people skills. Only magical people that know that Harry is alive are Lily, James, Sirius, Remus, Dumbledore, and McGonagall.
tl;dr: wbwl tropes, but put them together less stupidly. Everyone thinks Harry is dead and the Potters have been recluses in the mean time while trying to find where Voldemort has hidden himself and how he survived. The first reunion happens at the end of first year in the Hospital wing. They have an actually touching reunion because the reason they had to stay away makes fucking sense.