Well, there is no problem then. You must now use your new name for any immigration or non immigration forms.
Fill up an NTL application with your new name, and in any columns asking for your previous name, put your previous name, simple.
Send them your deed poll, and any other documents that you think will establish a link between your old and new names.
Just write a simple covering letter that you have changed your name for such and such date. Again, write a SIMPLE LETTER, it does not have to do any explaining or telling a story, like immigrants usually do.
hope this helps
