Thanks all, I wasn't really worried about navigating the subway as I am very comfortable doing that, it was more the hassle with the bags. Appreciate the advice and think I will probably go with the tube as we're both fit and our bags won't be heavy.

As a follow up question, on the return trip to Heathrow on Sunday, I'm considering taking the tube back but I can't seem to find what time the Piccadilly line starts to run on Sunday at the Transport for London page. Maybe I'm just missing it. Would it be running by 5 AM?

Thanks,
AGS