When I was young, you were looking at a nine month ban from competing for the new club, unless you had moved to a new part of the country. It seems it's now just 4 months, and maybe less if nobody objects. Read this page and look at the guide on the right.
http://www.englandathletics.org/page.asp?section=454§ionTitle=Club+Affiliation
However, we now have the "Higher Competition Athlete" rule in the British League to help folks like you. Google it. You can join a club in the British Athletic League and run for them in order to get a higher level of competition, without leaving your old club. This is the way to go if you like your local club, have a coach etc. Look at the BAL clubs nearby (www.bal.org.uk) and call a few of them up, as there's no point registering for one that doesn't need you in matches. BAL fixtures can be all over the country.
Roughly where in the UK are you, what are your events and PBs? Round it off a bit to avoid death threats from the geriatric club secretaries who frequent these boards looking for possible defectors ;-)