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I'd vote for Winchester too - I live there! Only 1 hour by train from London Waterloo. The ancient capital of England. The cathedral, the college and don't forget the chapel and hospital of St Cross
I vote for Hampton Court too. Only don't make the mistake we did: we went there by boat on the Thames but arrived so late that we had to rush through. However, returning by boat on the Thames would be neat (relaxing & scenic).