The bridge is still closed (see https://www.southwark.gov.uk/transport-and-roads/highways-defects-and-improvement-maintenance/bridges-subways-and-walls?chapter=3)
I don't believe there's a signed alternative route, but it is possible to use other footpaths south of the bridge in the wood to bypass it. This doesn't have a significant impact on the Green Chain Walk - you can walk down the east-side of the ex-railway cutting rather than the west. If anything this is a nicer walk.

I’m afraid I’m not sure. I mapped it a while back now and can’t remember. It looks mapped outdoors and the park has no gates so I suspect so.

Hi JodaStephen, thanks for pointing this out. Must have happened by accident when editing when out-and-about. It's all corrected now along with a few other minor improvements.