Beachcombing on the Thames

It's called mudlarking

Apparently you need a permit

http://www.standard.co.uk/news/london/mudlarks-on-thames-told-to-get-32-permits-to-save-heritage-a3412266.html

https://www.pla.co.uk/Environment/Thames-foreshore-access-including-metal-detecting-searching-and-digging

Called 'mudlarking' here, the main beach areas are around the Millenium Bridge, and there are a couple of easy access points on both sides of the Thames. The beaches are generally clean but you might come across small areas of very fine, wet mud, which you may or may not be able to navigate around. You could take a couple of large bottles of water to wash your footware down after, and i'd take some antibac hand gel also, or similar. There is a small, very muddy beach by Parliment, access not easy, and i'm pretty sure the beach buy Tower Bridge / Tower of London is closed except for a couple of days a year and then only to mudlarking clubs due to the fact they regularly find things of historical significance there. Have fun.

Permits: yes if your doing it properly with metal detectors etc. but just wandering along and scratching about on the surface, no one will care, and people are there casually looking all the time.

Be aware if the water level and your route out, it rises faster than you might think.

Not sure there is any intent by the OP to do digging but any digging requires a permit from the Port of London, starters only get a standard licence allowing to go down 7.5cm, and any objects of archaeological interest need to be reported Portable Antiquities Scheme at the Museum of London. There has been something of a crackdown on it in recent years.

From one of the pages given by ramblin' on:

Where on the Thames foreshore can I dig, search or use a metal detector?

The Thames foreshore permit is only valid for certain locations west of the Thames Barrier up to Teddington. Searching is not allowed east of the Barrier. All searching, digging, scraping or the removal of any items is strictly prohibited at Queenhithe Dock, Brunel’s Great Eastern Slipway, Tower of London and an area in Greenwich. (locations indicated in red on the plans below)

The City of London pages likewise have a warning to stay away from the beach area near the Tower of London.

I've been planning to do a bit of mudlarking myself. Look at London Walks and see what they have to say.