It won't be a problem. In the UK, your "legal name" is your assumed name, which has nothing to do with your birth certificate, your passport, or what government departments have you on record as. As long as you have assumed the name Joe (or intend to) and it hasn't been done to avoid debt or deceive, your legal name is now Joe.
If you had made a major name change (John to Mohammed?), I'd recommend getting a meaningless £40 deed poll from one of the private companies who provide them, because so many customer service people and small companies still believe they mean something.
If you want to update your name with government departments, they each ask for different "proof" of the name change, depending on their own (silly) rules. See: