I used to live around Wilmslow, and subsequently in Manchester. (I live in SW London now so know Richmond well too.)
When you say three times a week to London, do you mean there and back in a day? Wilmslow's on the west coast main line and there are usually a reasonable number of trains. It takes about 2 to 2.5 hours to Euston, depending on the train. As train journeys go it's alright -- it's fast and those Pendolinos that used to be Virgin Trains are comfortable especially if you can spring for first class. But it's expensive. You'd probably be looking at ~£140 each way unless you luck out and can travel at really weird times.
On Wilmslow itself... It's... fine... but I wouldn't describe it or its surrounding area as beautiful, particularly. It's a somewhat upmarket commuter town. Largely mid-century and later housing estates, not much period housing. I'm not trying to do it down but if you're expecting something like Ribble Valley/rolling hills type countryside, or indeed Richmond-esque period town prettiness, you're going to be disappointed I'm afraid. Knutsford is prettier in itself, although it's not especially well connected for London.
Driving to Withington from Wilmslow at school rush-hour time would be absolutely hellish IMO. The A34 looks handy and it often is but it also gets snarled up very frequently at rush-hour. A 20-minute journey could easily become well over an hour (I know this from bitter experience because I used to work in Stockport). Don't underestimate how much vehicular traffic there is in that part of the world; many more people drive everywhere because the public transport still isn't up to much compared to London.
The nearest thing to Richmond in that part of the world is probably Didsbury in Manchester, or perhaps Chorlton. Didsbury in particular would be handy for Withington and/or Fallowfield; you might be able to walk to the school.