Oundle is lovely lovely lovely I went to Oundle School and parents aren't far away and I love it.
Small but has a lot in it for its size and I find everyone so friendly walking around, people smile and are charming (I once left a phone in a coffee shop and the lady ran after me to return it, it's that sort of place). Thanks to the school you've got amazing amneities, Top quality gyms, olympic pool, art gallery, theatre, hockey clubs, rugby (v good) - it feels like a miniature oxford, very this style with the boarding school.
Not sure what Oundle the others have visited above but the school is not separate to the town it is almost the town very much like a mini Oxford in this regard - probably one of the most integrated schools in its locale in the UK. In regard Prince George, there is zero confirm but the security detail is minimal and very hidden. It was at Eton and Marlborough for Will, Harry and Eugenie. It will have zero bearing on your life if it comes to pass. The towns around are pretty awful. We use Cambridge if we want a bit of culture NOT PETERBOROUGH. But mostly just London as its a quick 47 minute zip into the capital and frequent trains.
It's also younger crowd than you'd expect thanks to the school and the teacher population so I find it more sort of liberal in a way not just a town full of retirees which I think Stamford can be a bit.
All in all you've got a great place. Personally I have come to prefer it to Stamford, but Stamford is lovely. Oakham I think is a step down (the development has been poorly managed), Uppingham is nice but think Oundle nicer; the villages around them are beautiful and the drive between Oundle and Uppingham through the valleys and undulating countryside is one of England's best - recommend for walks.
Good luck.