On the same topic, many people reckon that if you have luggage (and perhaps even if you don't), the change between the Piccadilly Line and the Jubilee Line at Green Park (in either direction) is best done by:-
This should, I think, be step-free and it avoids both a fairly long walk through the walkway tunnels on the offical connection route and the vagaries of the slow and often unreliable lifts that are needed to change levels along the official route.
For those who want to see the detail,
this diagram may help.
An alternative route between Heathrow and Canning Town is to do the standard cross-platform change from the Piccadilly Line to the District Line at Barons Court, and then using the escalators or lifts from the District Line to the Jubilee Line at Westminster. One more change, but rather less angst at either station.
Rumours that Mornington Crescent has anything to do with this post are malicious and unfounded.