People tend to check in during these times:
PAD-->Bristol Temple Meads on a Sunday afternoon. The busiest train I have ever been on.
To save money to Heathrow use the Heathrow Connect
Platform numbers come up on the National Rail iPhone app way before the station departure boards.
Sometimes you can piggyback on the Heathrow Express wifi while your train is at the platform.
Heathrow Express is the best way to London!
No wifi on trains. not even in 1st class :(