Take the train into London, find a hotel near the tube line to Heathrow
and take the tube out next morning. You'll be travelling against the
flow of commuters so it should be OK - certainly after the first few
stations.

It will work out cheaper than heading straight to Heathrow. Even if you
take the Heathrow Express there's not a lot of difference costwise -
the direct bus Gatwick-Heathrow is very expensive. Hotels at Heathrow
are as expensive as central London and all charge for a shuttle.