I've been off the bike for a year due to a couple of (non-cycling related) operations, but Southwark Bridge has always been the easiest bridge for me to get to.
So back on the bike today, I cycled north over Southwark Bridge today at 06.45 - the cycle lane was all clear, with the only issue being that the traffic is filtered briefly into one lane just ahead of the lights (but still 3 lanes at the lights I think).
Seemed perfectly straightforward to me ... but didn't notice anything on the southbound route, so maybe that is worse ...