Actually, we do, primarily under the controversial Digital Economy Act that was pushed through just before the last general election.
The laws aren't widely applied yet in the way that they apparently have been in France, but there has been some fairly high profile threatening going on all the same.
In other words, while I'm certainly not equating the situation in the two countries and the seriousness or otherwise of the threat to pirates right now, it's probably not fair to totally discount the effect of anti-piracy rules in the UK either.