Hi Suzie,
There is a conflict in these statements -
"My mum and her partner would like to purchase a house outright for me."
"My mum would like the property to be put in her or my name but her partner doesn’t want this. He would like to be joint owner of the property with my mum and said that he would leave his half of the property to me in his will."
So they're not buying it "outright" for you at all; they're buying it for themselves with the stated intention that they will leave it to you when they die. This brings all sorts of issues into play, particularly in relation to Inheritance Tax. If they give you the money to buy it outright yourself, that is a gift which is a Potential Exempt Transfer (PET) and as long as they live 7 years, that's an end of the matter. If they buy it in their own names, if they leave their shares to each other they will potentially incur IHT (a common misconception is that "partners" are treated as married couples for IHT. They're not) and if they leave it to you, the property will form part of their estates and potentially incur IHT on the value at the date of death (not the price they pay now).
I'm not sure why they want to do it this way but if they want to give you a property "outright", it should be your name on the title and yours alone.