I can't talk about schools but I'd say on a more 'general' idea:
Sleaford is crap, avoid. I literally cannot think of anything nice to say.
Lincoln is lovely, although always used to suffer a bit being 'inaccessible' but new bypass and more trains to London has really opened things up. Plus near the beauitful Wolds, and some nice parts of the coast.
Bourne is nice enough but just... small. Bourne Woods is lovely for a walk around, really nice leisure centre nearby, but A15 up to Lincoln or down to Peterborough if you want a city or Stamford not too far.
Newark is okay, but not really that much there. But it does have great road and rail links (A1/ECML).
Downham Market is miles from all those others as other PPs point out. It's alright from the few times I've visited but can't really comment much more.
Peterborough has some lively villages around it, but then so do some of the bigger places (Newark, Grantham, Oakham). But again it's what you are used to - do you want a small village with nothing much and constantly relying on car or a bigger city with a bus that is more than once an hour ?
Peterborough itself has some really awful areas but also some lovely areas . But it's a big commuter town given trains to London are less than an hour. The City Centre has had a bit of money spent on developing it, there is a nice big Waitrose right next to the station.
Have you looked at Grantham, Stamford, Oakham as other options?