Anything further West than Brighton WILL be more than an hour drive to Hailsham more times than not whatever the maps tell you - so I think you should prob rule out Worthing, Shoreham etc because of this.
Eastbourne really is your ideal town for non-driver, easy commute to Hailsham and range of good schools. Meads, Old Town and Upperton are the better areas. You'd prob get a VERY nice 4 bed detached in Old Town for your budget. Eastbourne can be fully self-sufficient, everything is walkable and has mainline train. Excellent art gallery, also theatres, and the seafront.
Lewes is prob the most desirable town- lots of lovely characterful businesses and people, and a better commute to London. You pay a lot for housing because of this. It's the one everyone bee-lines for - if the wider environs is more important than the house this is your place.
Uckfield is a good town, big enough to keep itself going. It has a trainline up to London but nothing out to Lewes, Eastbourne or Brighton. Its main town area is essentially one long high street. If you lived in one of the other towns, you wouldn't otherwise go there necessarily, but it keeps prices more affordable. A lot of development around Uckfield is going on - bit like Hailsham. Hailsham to Uckfield is car/bus only and can be a very congested journey at rush hour despite not being too far.
Bexhill (out East) is fairly similar to Eastbourne, but with a worse train journey up to London. It also has a great gallery/arts centre (De la warr pavilion).
Personally, I'd pick Eastbourne. Seaside, nice selection of cafes, good stuff going on in the town. Good private schools (if you want).
You're definitely looking at older populations in all these towns, and it is really very white.
But I like it here. I live in Hailsham - but prob tbh for your money you can afford somewhere with a bit more character than here :)