Hi Sadler, Turkish banks are not too happy about you taking your business elsewhere and are very slow to transfer services. Your best bet would be to close your existing account completely and cancel your DD's. Then open a new account at the bank of your choice and set up new DD's.
You will need your passports, a couple of current bills for your UK address and your Turkish tax number. You will be asked to sign a statement to enable either of you to access the account with just one signature. You will also need to register a mobile phone number with your bank for internet banking. Ask them to set it all up for you and ask for Debit cards. You will also need to take a bill for each of the DD's you want to set up.
I can recommend TEB, we have used them for over 4 years and been happy with their service, they are also one of the banks who will pay water bills on DD.