Richings park has no primary school but it’s in Bucks. Children who live there mostly go to Iver Infants (in Iver village proper) , I have friends in richings Park who are very happy with Iver infants: they commute and say the Iver Infants wraparound care is excellent as they have early dropoff from 7.15 and pickup at 6pm I think.
Iver Heath Infants is great but very oversubscribed: even some Iver Heath residents can’t get in if too far from school.
There is a primary Free school in West Drayton linked to the very popular church school (?st matthews) which is open access and driveable from Richings Park but it’s not very easy to commute from there so probably not suitable.
Next nearest to Richings Park is Parlaunt Park school in Langley, it gets some mixed reviews from Ofsted but has a great big field and a very cosmopolitan intake which gives a very inclusive feel, and it’s walking distance from Langley Station with parking nearby so you could drop at school then walk to the station. It also has the advantage that Parlaunt Park is a formal feeder to the oversubscribed Langley Academy comprehensive, which as a Bucks resident you’d not otherwise have any chance of a place in.
Prep wise if you want that, there is St Bernard’s Prep in Langley (linked to the catholic system but you don’t need to be catholic) or Long Close school which is by Upton Court Park and has a good reputation. If you have girls, Eton End is a lovely prep and they take boys in the nursery and infants section.
Farnham Royal has Dair House and others but that’s harder to commute to/from unless you got a nanny or au pair, a bit of a shlep for a daily school from Iver.
Langley Hall is a state free school whose senior site is in Langley centre, but their junior site is a mile away in Middlegreen. They offer a minibus service from Richings park for a tidy fee, but they have 800 pupils, a split site, and bus their senior kids across a main road for lunch with no outside school space. They are a popular school though and local opinion seems divided about whether they are brilliant or out to make cash! Certainly it gets complex, as a mum of three I avoided them as the kids would have been permanently split between the two sites due to their ages. They have a private nursery called wellingtons that was more expensive than the ones in Surrey and tries hard to imply it’s a feeder by calling itself “wellingtons for langley hall”. They were told a few years ago by the Dept of Education that it was illegal to have a private nursery as a feeder for a state school but they have retained the name because the executive head of the school owns the nursery and that is not illegal...
they do have lots of music and drama there, and the junior site has a small swimming pool. Depends on how much you like minibuses really. .. it could definitely be an option for you to look at.