London Bridge has never permanently closed in its 2,000-year history at this crossing. The current bridge (1973) and its predecessors have maintained continuous public access—day and night, year-round—with only brief, planned interruptions for:
When Rennie's 1831 bridge was replaced, engineers built the new box-girder bridge alongside the old one, then swung traffic over in stages. The medieval bridge (1209) was similarly replaced piecemeal. At no point was the Thames uncrossable at this point.
The current bridge has wide, lit footways on both sides, open continuously. It's a critical link in the Thames Path and a direct route between the City and Southwark (Borough Market, South Bank).
So if you're in London at 3 AM and need to cross the Thames at the historic center—London Bridge is open.