Officially, it is called a "Transfer of equity". Not sure why you need to know?
You will first of all need to approach your Mortgage Lender and explain you want to do a Transfer of Equity. They will then send you the appropriate application Form.
If they consent, you will need to instruct a local Conveyancing Solicitor to deal with the necessary Transfer and registration of it at the Land Registry. Your Mortgage Lender will also issue a new Mortgage Offer in your joint names.
It normally takes approx. 6-7 weeks from when you receive consent from your Lender.
You won't have CGT to pay.
I hope this helps. If so, I would be grateful if you could rate my answer.
If I have helped, I would be grateful if you could rate my answer.