I'd not recommend the Rotherhithe Tunnel at *all*. I've ridden it once (very late at night): I took the road, and the traffic was OK (I expect it's vile at rush hour though), but the air is *awful*. I was light-headed when I got out, and had a headache the next day of the sort I usually get if I get drunk and have a sneaky cigarette (I don't smoke as a rule). I think taking it daily would be actively bad for you.
doop, of this parish, does Bermondsey-C Wharf daily. He goes over Tower Bridge, along the first bit of Cable St, then right down Dock St, left onto the Highway, left again onto Butcher Row, right onto the bike lane through the park & over the bridge-in-the-sky (over Limehouse Link entrance). That spits you out onto Horseferry Rd (one-way eastbound). L at the end onto Narrow St. Right where the buses go just before the Westferry DLR station junction, then right onto Westferry Road and UP the road in the middle of the road (as it were) to Westferry Circus.
(Er. Can do Bikely route if that would be helpful!)
He says that Cable St is full of lemming-pedestrians on the bike path eastbound. It's better westbound because you can take the road with the traffic. The Hwy is busy (sometimes very busy), though.