I know that I coming to this rather late but we went through the exact same thing about seven years ago when we moved down from Manchester to Peterborough.
When it comes to independent schools, nobody here has mentioned Oakham yet. We found it to be an excellent school, they also do a mixture of day, flexi-boarding and full boarding which is great so that your DC can mix and match what they want to do.
Also, it's easy to get from Peterborough to Oakham by train and you'll see day pupils in the middle and upper school doing that so you won't have to drive them there and back.
When it comes to Peterborough, as others have said, you don't need to live right in the city centre as there are some good bus routes that easily get you from the suburbs into the bus station which is a 3 minute walk from the train station.
When it comes to primary schools, the better ones tend to be out in the small villages that surround Peterborough but there are plenty of fine primary schools in Peterborough itself. Although, there are also some truly bad primary schools as well.
Without naming names, in Peterborough it seems that the better the area is, generally speaking, the better the primary school is.
If you do want somewhere near the centre then I'm told that All Saints CofE is very good.
With secondary schools there are certainly some to avoid, such as the Queen Katherine Academy (that one has a long history of failing, it was so bad they've had to rename it a couple of times as it had such a bad reputation under its previous names).
Apart from Kings, Hampton is quite a good school and you might also want to consider Jack Hunt or Bushfield as well.