Well, the answers may change now anyway as the regulations are different in 2021.
If you have a PfCO or Operational Authorisation, then you may or may not be able to fly here depending on what your OA actually says. Unless you have an OSC allowing you to fly close to people, then the main issues will be the standard separation distances which are 30m during TOAL and 50m during flight, under UKPDRA01 (see CAP722 for details).
For any flight to the west of the bridge, this is in the City of London restricted area where you need to apply for special permission to fly a drone. Unless you’re a commercial operator, you’ve got pretty much no chance of getting permission, due to the risk assessment details required.
To the east things are very different. You are outside the restricted area, and in theory helicopter flights should not come below 500 feet. You can fly a small drone in Open A1 under VLOS up to 400 ft in this area with no required separation distances, the same as every other unrestricted area in the UK. The chances of flying under Open A2 are realistically nil as there won’t be anywhere you can guarantee 50m horizontal separation to uninvolved people.
I don’t know whether the PLA or local authorities have any byelaws in place.
However, the reason you see many negative responses above is because it really is not ok to just rock up and hang your hat on a simplistic view of the legalities. This is busy, congested airspace, and there may well be helicopters flying below the normal floor level, especially police. VLOS for approaching aircraft is restricted by nearby buildings.
Regardless of separation distances, you’re still legally required not to endanger people or property, and to be able to cease the flight if there is any risk to aircraft, people, environment, animals or property. It’s for you to explain how you can do that, not for anyone else.
You can also expect and should not be surprised if the police stop you regardless of the law. It’s a high-profile location with major security concerns. “Some bloke on the internet said it was ok” will get you nowhere.