Yes and no.
There is no formal test and licence for the activity itself, no ‘drivers licence for canoe’ as such. Anyone of any age can legally jump in a boat and start paddling. But you will need to pay for a licence or pass to gain access to most, if not all, inland waters of the UK. This will depend on who owns and/ or operates that particular waterway; for example the Environment Agency or Canal & Rivers Trust. But the easiest (and cheapest) method is to become a member of British Canoeing, which includes a blanket annual ‘Waterways Licence’ to paddle on thousands of miles of waterways. In a practical sense this will cover the vast majority of decent places to paddle in the country. It costs £45 per year for an adult at time of writing, which represents a huge saving when you consider temporary licences from the individual waterways themselves.
While getting formal training isn’t mandatory, it is certainly recommended if you want more enjoyment and safety from the hobby. Training courses will familiarise you with the equipment and terminology, techniques for moving and stopping, and for entering and exiting the boat. Safety and rescue will be a significant part of any professional training.
You can find training courses through your local canoe club, leisure centre, or British Canoeing which has a database of registered and regulated instructors.
Getting an annual Waterways Licence from British Canoeing isn’t the only option. You could get a temporary licence or pass from the waterway you are visiting, such as from a lock-keeper on the River Thames. This might be suitable for someone who is just on holiday for example, and not looking to paddle anywhere the rest of the year. But regular paddlers should consider getting the annual Waterways Licence.
I keep my British Canoeing membership card lashed to my PFD, so it is always available to hand if for instance a lock-keeper asks (which does happen).
WHAT ABOUT THE SEA?
There are no licences or passes for coastal paddling. There might be certain places you can’t launch or land, or have to pay to do so. But there isn’t any agency or trust collecting dues and issuing licences for access to the coast in general, and you wouldn’t need British Canoeing membership for this (although most sea kayakers are members anyway).
While all forms of kayaking present risk and danger, the sea is a particularly formidable and dangerous beast. Sea kayaking should not be attempted without any formal basic training from an instructor.