Forum Moderators: open
Personally, dragging my luggage onto the tube during rush hour wasn't high on the priority list of my London sightseeing itinerary ;)
I'd leave a good hour for the trip.
But seriously, the fastest way is Heathrow Express to Paddington, taxi Paddington to the Thistle.
And killroy: I'd find out which airport you're landing at. There's more than one, and they're in very different directions from London.
PS. Hope to see you on the train Brett.