It's fairly busy but mostly with river ferries, barges, recreational boats, etc which aren't too tall and can fit under most bridges. Taking the Tower Bridge example, vessels up to 9 metres tall can fit under it. If you're taller than that, the bridge must be raised on request[1]. I'd imagine that would be the minimum requirement for any bridge built east of Tower Bridge. Currently the only bridge over the Thames east of Tower Bridge is the Dartford Crossing, which has a 61 meter high deck.

[1] By law, river traffic takes priority over road traffic, but it's raised pretty infrequently now days.