OK ok can we please stop with the Old Hastings mush.
It is referred to as Hastings Old Town, The Old Town and original residents as Old Towners.
Property comes up rarely and can be pricey, it is a saught after area. I lived there from 1992-2000ish so any questions please ask.
There is also Clive Vale which backs off the Old Town, it is an average area but the secondary schools served by it at present are awful. However, if you aren't careful the local estate agents will sell you Clive Vale as The Old Town and local people will laugh at you if you try and convince them you live in the old town 
Oh and the school catchement areas are the same as Clive Vale, good primary poor secondary.
Although that is the same for the whole Hastings area at present, but new Academys have been announced.
I now live in Battle which is a beautiful market town between Hastings and Eastbourne (inland not on the coast). It has excellent primary and secondary state schools and plenty of private schools to choose from locally.
Bexhill is on the coast and cheaper than Battle. It is locally considered God's waiting room as it attracts a large amount of old people, but it is a pleasant area with some nice old buildings. Again though state schooling (unless you are catholic) is poor.
Eastbourne is lovely (my brother lives there) but as with all towns you need to pick your area carefully. Schooling is marginally better than in Hastings or Bexhill, but these things change quickly so checking out up to date info is a must.
Can I ask where in London you are now and what it is you are looking for and what you 100% want to avoid (apart from Chavs and wierdos)