Nope, I never thought I had never heard of this "mudlarking" lark either, but it turns out I have, just through a different name!
Mudlarking is essentially just scavenging or treasure hunting, and whilst you don't have a beach to use for said treasure hunting in London you do have the River Thames. The Thames has been at the center of city life in our capital for hundreds of years, and with it being the main method of transportation for the pre-car era there is a lot of "treasure"to be found along the banks.
So of you fancy yourself as a bit of a treasure hunter you've got 95 miles of the tidal Thames to have a scavenge through. There are however some rules to go with mudlarking. You are allowed surface finds without needing a license. This means that you are not able to overturn even one rock without a license. You can apply for a license here. Obviously safety comes into play as the Thames tide comes in twice a day and rises to above 7 meters, so unless you're quite tall you would be in quite a bit of trouble! Just make sure you have a look at the times that the tides come in and out before you start mudlarking.
If you do manage to find anything on the shore then the protocol is to declare it to the British Museum so that it can be enjoyed by everyone. I'm 99% sure that you won't find any gold or diamonds along the shore, but if you do, just remember who told you about the thing in the first place!