At the end of January, I'm flying into Heathrow on a Friday morning for and must be in Birmingham that evening and I'm trying to figure out what I can squeeze in, also need to store bags while we sight see.

My tentative plan
Arrive at 6:50AM on Delta - LHR T4
8:00AM? Heathrow Express to Paddington
8:20AM cab or tube to Euston Station
8:30AM to 2:30PM sight see

How crazy and packed is the tube on a weekday morning? Would we be sitting in traffic if we took a cab? Also would 6 hours be enough time to take in most of the big sites?

Thanks