Crossing London Bridge costs nothing. It is a free public highway (part of the A3 road) maintained by the City of London Corporation and Transport for London. No toll, congestion charge (unless entering the zone), or access fee applies to pedestrians, cyclists, or vehicles using the bridge itself.
The original medieval London Bridge (1209-1831) did charge tolls—a major revenue source for bridge maintenance and the City. The "bridge toll" concept persists in cultural memory, but the current bridge (opened 1973, replacing 1831 Rennie bridge) has always been toll-free.
Walk, cycle, drive, or bus across London Bridge freely—just don't confuse it with Tower Bridge (the photogenic bascule bridge downstream) or the London Bridge Experience (paid tourist attraction in the bridge's vaults).