Polly - forms AP1, TR1 and ID1 are the correct forms needed to transfer the legal ownership and apply to update the register
Form TR1 is the legal deed and the linked guidance explains how to complete it, inc the consideration and fees payable
Speak to the mortgage provider to establish what they require. You may find that they insist on your using a conveyancer
Registering land or property with HM Land Registry: Change the registered owner name - GOV.UK