Can anyone recommend a seaside town which has a scenic walk nearby that you can get to by train from London in under 3 hours?
Much of Britain's south coast is within a 3-hour train ride from London. Brighton, for instance, is one hour by direct train from London Bridge station and has the fabulous Royal Pavilion, in addition to the usual seaside frivolities. www.buses.co.uk can run you out to see the white cliffs at Beachy Head. Brighton is covered in Rick's England book, but not his Great Britain or London books.
Dover is an hour and a half by train. Might these be of interest to you? http://www.theguardian.com/travel/2007/may/02/top10.walkingholidays
Just about anywhere from Hampshire to Kent will be within your time limit. It depends on what you are interested in. There are towns that are full of history (Brighton, Hastings, Chichester) and those that cater to family holidays - Eastbourne, Bournemouth, Ramsgate. Not many have sandy beaches. Eastbourne, Brighton and Dover offer great walks on the cliffs (Seven Sisters, white cliffs of Dover, Beachey Head) and there are some nice walks outside the towns but a car is needed.