You can cycle into and out of both Heathrow and Gatwick airports (I've cycled out of both airports), although finding an enjoyable cycle route might be a challenge, depending on where you want to go. You can also cycle from either airport to a train station and take your bike with you on a train to a more cycle friendly environment.
Sustrans (www.sustrans.org.uk) publishes maps of various long distance cycle routes throughout the UK, including a few that start in or near London. These routes go to great lengths to route cyclists away from auto traffic. Some people think they go too far in avoiding auto traffic, but they're a good way to get through the most crowded parts of the UK (like London) without riding on really unpleasant, busy roads.