My usual route to/from Paddington is to take the cycle route that follows the minor roads to the south of Marylebone Road and Euston Road and then go down Gray's Inn Road.
You could also try to pick up the canal towpath near to the station (it goes through a tunnel so you have to pick it up after the tunnel). That would get you at least as far as Kings Cross, maybe further.
The
tfl.gov.uk journey planner is your best bet. I hope that link works but it may send you into a loop where the page just keeps refreshing.
I would seriously avoid taking the tube. Maybe on a quiet Sunday morning but not otherwise. I would far rather take my chances on the road than try manhandling a bike and luggage on the Circle Line (Hammersmith and City might be a bit better).