I've had to get from Canning Town to LHR in the morning rush hour after I got a call to say the flight was cancelled and it was hours to the next one.

Even with luggage, the only way I would consider this is the Jubilee line from Canning Town to Baker Street. Even if you have to wait a couple of trains at Canning Town before you get on, once you're on then you just stay there all the way to Baker Street where it's an easy connection and the Bakerloo line is going away from the centre rather than towards it. If the escalators at Canning Town and Paddington are too much for your luggage then I'd just plan to miss the connection and take it gently. Depending on how long it takes to get through immigration & to get your luggage, if it's clear you won't make LHR in time then it might be worth asking if BA at LCY will re-book you.