Regardless of what the weather is like on land at either end of the bridge, the weather on the Tower Bridge sucks. Bring a jacket if you intend to cross it on foot.

That being said there's lots of cool stuff on either end. Bring lots of limeybucks and plan to spend the day in the vicinity.

Be sure to check out the HMS Belfast, pictured below...

This was once a port where tall ships could unload their cargo out of the wind and rain. Now it's a shopping mall with a weird metal sculpture. To get here keep walking away from the bridge and past the Belfast till you see this place. Passing through and out the back will take you to the London Dungeon theme park...