First of all plan your letter in advance so that you make sure all the relevant information is included. You will need to gather the information required, for example you will need the bank account number and sort code of your account and probably the address of at least your old branch and maybe your new one. I would think they will be able to track down the relevant branch from a street name and they will certainly be able to find out the new Sort Code for you if required.
Once you start writing keep it as concise as possible, there really is no point in repeating yourself so keep to the facts and omit any unnecessary waffle. If you know your bank manager by name you can address it to him as Dear [Mr X] and end with Yours Sincerely, otherwise stick to Dear Sir/Madam and Yours Faithfully but you should remember that this is a business letter so these are expected.
Then in the body of the letter you simply need to say:
I would like to transfer my account (insert account number and sort code) based at your (insert address of branch) branch to your branch at (insert address of new branch).
It may also be possible to do this in person at your branch or even over the telephone.
Once you start writing keep it as concise as possible, there really is no point in repeating yourself so keep to the facts and omit any unnecessary waffle. If you know your bank manager by name you can address it to him as Dear [Mr X] and end with Yours Sincerely, otherwise stick to Dear Sir/Madam and Yours Faithfully but you should remember that this is a business letter so these are expected.
Then in the body of the letter you simply need to say:
I would like to transfer my account (insert account number and sort code) based at your (insert address of branch) branch to your branch at (insert address of new branch).
It may also be possible to do this in person at your branch or even over the telephone.