Salisbury hospital is pretty good in my experience. They seemed to handle the pandemic pretty well anyway@Dr House
I would suggest the Hampshire, Wiltshire, East Dorset area.
Some of your criteria make it challenging re hospital access and schools.
It depends what your property expectations are but for example around the East Dorset area you can get good sized 2000-3000sqft+ properties in budget, still have 2 hospitals close by, good communication links (A31 / Regular 2hr trains to London Waterloo / Southampton Airport), good private schools etc etc.
There are pockets in the Hampshire/Wiltshire Ringwood - Salisbury - Winchester - Fair Oak - Northern Southampton regions too but hospital access isn't always as easy.
Great pies in MelbourneQuite like South Derbyshire myself, there are some nice little towns/villages around (Melbourne, Belper etc). You're 45 minutes from the Peak District and within an hours commute of Brum, Leicester, Nottingham, Derby so plenty of work options. Plus you can get to pretty much anywhere in England or Wales pretty easily.
Added bonus....if you nip up the A50 you can experience the Stoke Stink
Great pies in Melbourne
Would second Bath - anywhere in the south is in pretty easy reach even London more or less and a nice place in itself.
Jaywick
Another place I really like, the other half is from Glossop, so I spend a lot of time there. Really great spot nestled in the Peak, but with a 30min train into Manc. Got some infrastructure problems, but a really nice place. Only problem is the in-laws otherwise I'd be keen ;P
Any other suggestions? There seem to be nice houses around Cambridgeshire.