Ah! Good old river access!
No, there is no closed season as such, some rivers are paddleable all year whislt others are only open for a short period. For example, you can paddle the Medway between Tonbridge and the sea all year (as long as you buy a licence from the environment agency or are a member of the BCU) whereas the Mole nearby is can only be paddled for three months (March 15th - June 15th I think) and even then only with permission; I understand that this is a closed fishing season on the river, but since I know nothing about fishing I may be wrong.
In general (as I'm sure you know but I'll mention anyway to get some overseas support!) we only have access to 2% of inland waterways in England and Wales. Some limited further access is available by negotiation, however, unlike most countries, you cannot assume that you are allowed to paddle any particular river at all, whatever the season.
This link may be of interest to some;
www.riversaccess.org/