I wouldn't exactly say it's close to London, but the North Norfolk
Railway in Sheringham is doable in a day. It's about three hours each
way from Liverpool St, with a change at Norwich. The current
Sheringham platform at the end of the line is across the street from
the old station where the preserved railway starts. When I was there
a few years ago they were planning to rebuild the level crossing there
to reconnect them, but I don't know whether they actually did.