Within Europe, including Britain, Great Britain, UK, England and so on you should have a two year warranty courtesy of European legislation. Indeed, the warranty card supplied with the Note should say that it has two years cover (24 months) within Europe.
But that's by the by. I doubt the problem here is anything to do with a warranty related fault as it's unlikely that a hardware fault would result in a slow machine - either it will work or not. So the problem is most likely in the software configuration somewhere. I upgraded following a complete wipe of cache, data, Dalvik and there have been no problems with lag etc..