Originally Posted by Bloomsbury

Heathrow would be my choice. Jump in a cab to Paddington and get the Heathrow Express.

I'd say that if you have your HEX ticket in advance and time things right, you should be able to get from Hyde Park corner to T5 in approx 45 minutes with a fairly pleasant and trouble-free journey.

The tube would take you directly there (if you get a Heathrow train), cost less and could be quicker, but is likely to be more crowded and unpleasant (particularly if there's a lack of Heathrow trains).

I personally wouldn't bother with LCY as there's too much to go wrong at that time of day to get you there quicker and less stressed than Heathrow.