Originally Posted by
Mizter T
From County Hall it's easier to just walk across Westminster Bridge to Westminster Tube station and pick up the same first Jubilee line train there (rather that from Waterloo station). Also, the arrival time at Heathrow Central tube station is actually 06:36 not 06:45 - perhaps the latter timing takes into account the walk from the tube station to T1 (which isn't really that far).
If you go from Westminster it's better to take the District / Circle Line to either Hammersmith or Barrons Court from where it's a few feet walk to the adjacent platform for the Piccadilly Line to Heathrow.
At Green Park it's often easier with luggage to take the escalator up to the exit but rather than go through the ticket barriers turn round and descend to the line you want.