As long your deed poll was signed by you using your old name (so that the signature on it can be compared to the one that's currently on your bank's records) and independently witnessed, and you attend the bank in person (so that they can then get you to sign in both your old and new names, with a member of the bank's staff present to witness you doing so), there should be no reason why the bank won't accept your name change. e.g.
https://www.barclays.co.uk/help/your-details/change-name/
You should be able to change the name on your driving licence by completing form D1 (available from many Post Office counters or by ordering one here
https://www.gov.uk/dvlaforms ) and submitting it, together with your deed poll (or a certified copy thereof), together with your current licence to the address shown on the form:
https://www.gov.uk/change-name-driving-licence