Google (using it's Walking directions) reckons that it's 30 miles or so from Heathrow to Lee Valley, and only 35 if you go via central London, so if I was you, and had the time I'd go via the centre and at least cycle past a few of the classic tourist sites like Buckingham Palace, Westminster, the Tower of London, and Tower Bridge.
Hunting around I did find
this page which has some information and comments about cycling in and out of Heathrow.
I'm sure there are people who can make suggestions about routes. I'd offer to cycle out and meet you there, and take you through London, but I can't say I'm any particular expert in either of those directions. I live slightly South of central London, which is not either of the legs you'll be needing to travel.
I'm in awe of anyone who'll travel across the Atlantic and then kill themselves for several days cycling more or less non-stop across the country! Are you planning on taking any time to rest and get over jet lag, or are you just going to treat that as part of the experience?