Originally Posted by irmster
I wouldn't recommend the Piccadilly Circus option anytime soon. Pic line will be closed from Hyde Park Corner for some time yet, and if you get caught in one of the security alerts, you may have trouble getting back.
I would be prepared to take the risk (and the risk is not THAT large). I'm also not prepared to let bombers, security alerts and the like stop me from going about my daily life.
Get the Picadilly line to Hyde Park Corner, leave the station and pick up one of the many buses going up Piccadilly, or grab a cab the last mile or so to Piccadilly Circus. This will add 10 minutes or so to your journey.
In the unlikely event that Hyde Park Corner or the Piccadilly Line gets hit by a security alert when you need to get back to LHR, jump in a taxi to Paddington and take the Heathrow Express (cheaper and faster than a taxi all the way to LHR).
Enjoy your day out in London!