hi OP, this was me last year, I could move anywhere out of London, though budget was lower than yours. After being spoilt by living in nice london locations, I was very weary as to what could compare, but on the other hand it was fun travelling and looking at completely differnt places! ?Funnily enough I looked at most cities mentioned here - Harrogate, Brighton, Bath, Bristol, Cheltenham, also Oxford, York and Winchester!
It really depends what do you like London for? is it cultural stuff, the buzz and hte crowds, good transport, the parks??
Also important: would you need to visit London often once you move? As if not the case you are more spoilt for choice. I was shocked at train ticket prices, even IF trains are nice, but many trains are uncomfortable (why aer thrains to Sheffield and the North are SO overheated that you end up dehydrated?)
Winchester is really not great for a non-family setup! it's very small, snobby and only worth living there if you want schools and a good commute to london. There really is not much to do there at all, even though it's pretty and looks nice for a visit.
Bath I would recommend! My budget didn't quite stretch to a good size period two-bed walking distance to station. Very good theatre, shops, big park plus countryside nearby, Bristol not far for more shopping etc, and it's not oo small, plus 1h30min to london paddington on a nice train (not thatcheap of course).
I went for Bristol as it has the BEST choice of nice flats andthe prices are lower than Bath. For 350K I wouldn't go to Southville unless you want a big house. I'm very Clifton/edge of Redland biased, it's also near Univresity that gives it a lively edge (but not overcrowded). You aer near the Downs and Suspension Bridge that leads into a forest and a huge green estate as well, you get villagey feel in Clifton viillage which is vey charming and beats Winchester imo! Tp me brighton is jus ttoo crowded especially with hen/stags in the centre, you don't get that around here. There aer a couple of good theatres in Bristol (Old Vic), and shopping is good.
You can get a very nice 3bed period flat for 350K in Clifton. The only downside is that these nice areas are not by hte main station, yo need a connecting little train (about 20 min, twice an hour) to get to Temple Meads, so overall it's 2h20min to london with connections (the main train is 1h45min. of course you could take taxis or buses to main station (about 25min, but hugely frequent). Btw Clifton is much better for public trabsport (look at whiteladies road) with lots of buses into the centre and out of town, unlike South Bristol (Southville) so be aware of that
Bishopston is quite nice but it's a families place and a bit crowded, depends what you like!
Bath just has the edge on Bristol with shorter journey to london, more cultural stuff possibly, and being more compact, but then the prices ARE high, 300-320 for a 2bed.
Harrogate has very beautiful properties and countryside but it's a bugger to get to from london with connecting train fron York being old and slow, but if yo uwant to stay put and mainly just go to York and Leeds then it's an option. It's too small for me though after London.