The situation varies depending on where you are in the United Kingdom.
In Scotland, since 2005, there is a general right of access to all land and water provided that rights are exercised responsibly. The Scottish Canoe Association produce an excellent publication, "Key Points of the Access Code for Paddlers", which gives a quick reference guide to situation since 2005. You can download a copy here.
England and Wales the situation is much more restrictive. In almost all
cases you can paddle legally on tidal waters, though you may need to
pay harbour dues; you can paddle on common law navigations such as the
Wye below Hay-on-Wye and the Severn below Pool Quay near Welshpool,
though a licence is needed below Stourport.
On most other waterways you will need a licence obtainable from the individual waterway authority (such as The Canal & Rivers Trust, The Environment Agency for the River Thames and the Basingstoke Canal Authority). Alternatively you can join Caoe England or canoe Wales. Among the many other benefits, membership includes a Canal and Rivers Trust and Environment Agency licence that enables you to paddle on many of Britain's canals, statutory navigations, and on some private waterways. The cost of an Adult Basic membership of Canoe England for 2013 is £37.00.
A more detailed list of the waterways covered by Canoe England membership is as follows:-
Locks and Tunnels
The British Waterways (now canal & River Trust) booklet " The waterways Code " includes the following advice - " Locks, weirs and sluices are dangerous. Don't stay aboard your craft in a filling or emptying lock - carry it around, or if it is too heavy, use lines to keep control." You should also be sensitive of the need to conserve scarce water resources on canals that experience this as a problem. It is a different matter on the Thames where the lock keepers are generally happy to take canoes through the locks.
The situation regarding canal tunnels is that BW has three classes of tunnels. Canoes are prohibited from using class 1 tunnels, may use class 2 tunnels subject to specific regulations and have unrestricted used of class 3 tunnels. See here for details of which tunnels are in which class. (if you are having dificulty wiewing this pdf file a text version is available here.)
BCU Rivers Advisory Service
The British Canoe Union, via the individual national bodies, has a network of River Advisors (covering England & Scotland but not Wales or Northern Ireland) who can provide information on any access issues and local voluntary agreements. Click the relevent country to get details including email addresses.England
Wales (no River Advisors)