I know that you've already turned down Peterborough as being too far away, but if you/your DH go on LNER then Peterborough to Kings Cross is 50 minutes.
"Max budget is £525k. Need a 3-4-bed house"
OK, no problem. What are you looking for?
A new build? Loads of those. For example this one for £500k; 4 bed, 2 bath, 181 sq metres, backing on to a lake. According to google maps, it's 6 miles from the railway station and takes 14 mins in the car - although obviously longer by bus
https://www.rightmove.co.uk/properties/146871272#/?channel=RES_BUY
or this one which is nearby on the same development, also for £500k:
https://www.rightmove.co.uk/properties/142065656#/?channel=RES_BUY
The local primary school in this area is very good and achieves above average results in the Key Stage 2 attainment figures (the year 6 SATs)
Or if you prefer more 1970s-1990s homes (which give you more space than new builds) then plenty of those as well, for example this is typical, a 4 bed detached with a nice sized garden for £490k. It's under 3 miles from the station and takes about 8 mins to drive there.
https://www.rightmove.co.uk/properties/131863205#/?channel=RES_BUY
Or 1930s homes? Here's a 1930s 4 bed detached with a good sized garden for £480k that is less than a mile from the railway station, so you can walk there in 15-20 mins. It is also literally less than 100m away from one of the top state secondary schools (the Kings School, Peterborough) not just in Peterborough, but the whole of East Anglia - at least, according to the Times & Sunday Times School League Tables 2024.
https://www.rightmove.co.uk/properties/147342293#/?channel=RES_BUY
Or this 1930s property. 4 bed, 3 bath, 161 sq metres on a plot of 0.4 acres (it has a huge garden) for £475k. It's 2 miles from the station or about 6 minutes:
https://www.rightmove.co.uk/properties/147016487#/?channel=RES_BUY
Or do you fancy a really old, stone built cottage, with sash windows, about 167 sq metres, 4 bed semi-detached with a reasonable sized garden? About 3 miles or 8 minutes from the station:
https://www.rightmove.co.uk/properties/144320675#/?channel=RES_BUY
There's quite a lot to choose from at that price point.
Kings School
and there are threads here about how to get into that school