Most of Winchester is lovely. It depends what you want really. I'm being a bit nit-picky with the downsides here. You have to consider what is important vs what you could compromise on.
Central Winchester, Hyde, Fulflood, St Cross - although not quite London prices, are heavily influenced by being within walking distance to the station.
Olly B and Badger Farm are fine, brisk walking distance to Kings.
The Romsey Rd is a nightmare during rush hour and school start/pick up times (so many sets of traffic lights) and the traffic can back up beyond the Pitt roundabout down to the M3 or towards Hursley.
Stockbridge Road and the roads off it are good too. However at pick up/drop off in and around Westgate can be a bit of a chore.
The roads adjacent to Peter Symonds (Cranworth, on the station side and Berweke on the other) get jammed with 4000 students heading to/from college and being picked up for their driving lessons!
St Giles Hill is good, but you can hear the motorway and the inbounds to Southampton airport go overhead.
Just be aware of the following:-
Some of the roads around River Park St Bede's school flooded last winter.
Because of the proposed development of Barton Farm, adjacent to the Andover Rd - there is a degree of uncertainty as to how much traffic this will generate and which roads will be turned into rat runs.
Some (much, sadly) of Stanmore has been sold to landlords to be rented out to students, so what would have been solid 3 or 4 bed family homes are now HMOs.
There are some lovely villages near to Winchester just check the school catchment map, it is a bit counter intuitive being almost a 'figure of 8'!