Agree with everything MustBe said, Clapham, Oakley, Sharnbrook all perfectly nice villages with various shops, pubs etc. Oakley probably my fav of those 3 as the prettiest, but it really depends what you're after, Clapham is the nearest to town and also probably the biggest, lots of new housing going up there. Harrold is perfectly nice in itself but further out, probably at least 30 mins drive in the morning to school (depending on traffic) as opposed to maybe 15-20 mins for the others? But you'll certainly get lots of Harpur Trust parents coming in from that kind of distance so you won't be alone. And if you don't mind being that far out, Lavendon and Turvey are also v nice?
As MustBe says, Biddenham and Bromham are probably the most popular 'villages' for Pilgrims/HT parents, I put that in commas because in my mind they aren't really distinct villages, they sort of run into the town and they have a bit of a generic-y, 'executive home' feel to them. But perfectly pleasant, very convenient for town (lots would commute to London from there) and plenty of choice of good quality housing particularly if you are after a more modern style house.
In terms of living in the town itself, the 'Castle Quarter' as the estate agents have taken to pretentiously calling the streets around Castle Road, is very nice, safe, lots of chi-chi fancy delis and cafes, very family friendly. Handy for town, your DD could walk to school and you have the parks, the embankment and restaurants/bars close by. But of course it is in the town itself so will have a different feel to the villages. Most of the housing around there is Victorian or 1930s era, which may or may not be your thing - either the grander streets around the embankment (many of which have been divided into flats or aren't residential any more) or the streets of terraces and semis further down Castle Road - lovely period houses but tend to have small gardens and no off-street parking, is that what you are after or do you more want the detached house with driveway and garage?
If you do want to live in the town itself you could also look at the streets around Bedford Park (I think the EAs refer to as 'Poet's Quarter') as that would be even closer to school, or Putnoe and Brickhill are fine although more modern style of housing and there are a few council estates there if that bothers you?