Have him put the house in your name before he dies and you will avoid any tax
Not necessarily true. He would need to stay alive for 7 years after giving OP the house, for her to avoid inheritance tax. If he dies before that 7 years is up, she'll have to pay some IT.
OP, neither of you should be putting money into a house you don't actually own. I don't understand how you can bequeath a property that you don't actually own either.
But you do need to change your will regarding the first house, if you want your child to inherit anything from you, until such time as you do actually own the second house. Then you can change your will again, to the original plan.
I used to see so many stories on another forum, where the mother or father's assets went to the other partner, who then remarried, died, and all assets went to the new spouse, with the children from the first marriage losing out.
Someone earlier in the thread asked why you would even think about splitting up, when arranging finances, that if you do consider it, your relationship is in trouble.
Well, actually, everybody should consider all possibilities, however unlikely they seem, or how strong their relationship is. It's prudent to do so, as many people here will have, at one time, never believed their partners would have an affair, only to find years down the line, that they're left for the OW.
So many relationships do end, so it's wise to plan for the worst, and hope for the best.
It's not a question of trust, or lack of it.