There is a website at www.gov.uk/register-to-vote where you can update your address on the electoral register to your new one.
It says:_
"Use this service to apply to register to vote or to:
update your name, address or other details on the electoral register
change your voting preferences, eg to vote in person or by post
change whether you’re on the open register
It usually takes about 5 minutes.
You may need the following, if you have them:
your National Insurance number
your passport if you’re a British citizen living abroad"