The Olympics are from late July to early August when half the UK is on annual holiday anyway.
Yes, Heathrow and much of London's infrstructure will be put under strain, but it's really just 3 weeks of peak activity. People will grumble and moan, but they will all know it's a short term one-off thing, be a bit more tolerant than usual and that everything will largely return to normal by late August.
Heathrow's see far worse when snow has been dumped on runways just before Xmas in recent years and swathes of flights have been cancelled. People will cope.