Salisbury is out in the back end of nowhere. Definitely not commutable to London for evenings out, etc.
I live in Chippenham, which is on the direct line into Paddington (we're 10 minutes on the same line from Bath and 20 from Bristol). Takes an hour to get into Paddington. We're also 4 miles from the M4 junction and can be in Manchester in 3 hours by car or change at Bristol to go up by train.
We regularly go into London for the day/evening. I recently took DS2 to see an evening showing of Frozen. We got the 11.30pm train home, but could have comfortably made the 10.30pm one. We went up at lunchtime and had a lovely day wandering around beforehand.
You'll comfortably get a 3 bed with a garden for 550.
Definitely don't move to Mitcham! I grew up in Wimbledon, and lots of my friends lived in Mitcham, so I spent a lot of time there. Grim sums it up, though it does meet your criteria for commuting/price.
Chippenham is a market town with a good sense of community, lovely nurseries, and excellent schools. John Coles Park is excellent and has Green Flag status.
We sort of ended up here after initially moving from London to Bath. But it suits our needs perfectly. Lots of families here are in a similar position. Didn't plan to move here, but now wouldn't leave.
Great shopping in Bath, Swindon and Bristol. More limited in Chippenham itself.
I found this property on the Rightmove Android app and wanted you to see it: www.rightmove.co.uk/properties/129842135
This one is a 5 minute walk from two excellent nurseries, 2 minutes from a great primary school and 5 minutes from a good one, about 7 minutes from both town and the train station.