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In order to transfer your property into joint names, it is a simple case of instructing a Conveyancing Solicitor to prepare a Land Registry Transfer document, which is signed by the 2 of you and then registered at the Land Registry.
The only possible tax which is payable on a Transfer is Stamp Duty, and this is only payable if you were paying more than £125,000 for your share or if you were assuming a debt (ie taking on any Mortgage already registered against the property) for £125,000, which you will not be. Hence no tax is payable if your partner is gifting you half the value of the property.
Your Solicitor is likely to charge you approx £250-£300 plus VAT for this work, and there will also be a Land Registry fee to pay, based on the value of the property and which will be anything from £40-£90.
I hope this assists and sets out the legal position.