Arriving at 8:00am means that you won't be in the arrivals area before 9:00am, so by the time you get to the train station you may as well get the ordinary train.

But rather than get the Tube northwards, get the 16 bus from outside Victoria which takes you virtually to the door of the Marriott

Have a play with http://journeyplanner.tfl.gov.uk

As for Tube on a Sunday, first thing to do is check if they are doing anything with the tracks that day / weekend

http://www.tfl.gov.uk/tfl/livetravel...e/default.html

Easiest way to LHR on a normal day BTW without using the HEX / HEC is the 16 bus back to Victoria, then Tube to Hammersmith or Barrons Court then walk a couple of yards across the platform and take the Tube to LHR