Thanks for your question. I will try to help.
As there is a mortgage on the property then only way that you would be able to transsfer the property in to your joint names would be if you were either able to repay the whole of the current mortgage from your private funds, or to obtain a mortgage offer in your joint names and then carry out a remortgage and transfer of equity.
However, in either case (and any case where two people get a mortgage to own a property together) you would be jointly and severally liable under the mortgage. This means that as far as the mortgage lender is concerned, you both owe the whole sum of the mortgage to them. So, they can choose to sue either or both of you for the whole amount in the event that you default.
However, once you have obtained a mortgage offer and transferred the property you may then make a declaration of trust, which is a deed sworn by you both in which you specify how the proceeds of sale of the property (ie. when it is eventually sold) are to be divided. This would specify that you own differing percentage interests in the property, which you selecting whatever interests you consider fair. It would also state that it is subject to any other responsibilities you both agree you should have.
A declaration of trust would be enforceable by you against your partner (and vice versa) but not against the mortgage lender.
If you are not able to obtain a mortgage offer in your joint names, then you can still execute a declaration of trust which states that although your partner is the legal owner of the registered title for the property, that you have a percentage interest in the equity of the property. This is somewhat less secure than actually being named on the registered title for the property however,.
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