The speed limit under Tower Bridge is unlimited.
Your craft must proceed at all times at a safe speed when navigating on the tidal Thames. Whilst there is no specific speed limit downstream of Wandsworth Bridge, your speed must be appropriate for the prevailing conditions, the size and type of your vessel, the tidal conditions, your location in the river and, importantly, the safety of others navigating in the area or working or lying alongside. You should err on the side of safety when determining your speed so that there is no risk of injury to persons or damage to property.
There is a speed limit of 8 knots in the following places:
above Wandsworth Bridge;
in all creeks linked to the tidal Thames;
off Southend (in the inshore area)
at Coryton and Thurrock when gas tankers are berthed
Even at 8 knots many vessels can create an unacceptable amount of wash and you may need to reduce speed further in the vicinity of piers, smaller craft or river works.
Hopefully, I have not broken any copyright laws with the above statement..
Take a look at http://www.portoflondon.co.uk/
for the full SP
Some more Pics
Dome, Bridge and Richmond Lock
You have to go through the lock when the tide is low as the authorities keep the water level higher in the upper reaches of the tidal end.