Quite simple then, it now belongs to you and not your mum.
There may be an issue if the value of the house when she gave it to you was more than £325,000 (or she had also made other gifts as well).
If this is the case then, if she dies within seven years of the gift then you will be primarily responsible for paying any IHT due on amounts above £325,000.
But there is a sliding scale which decreases over the years until you get to seven years.
Your mum is allowed to pass on up to £325k in the seven years before her death before IHT starts to get involved.
Here is a practical example. Let's say that the house was worth £425k (with no mortgage) when she gave it to you.
That was four years ago. In the very tragic circumstances that your mum passes away tomorrow then her entire estate passes equally to you and any siblings that you might have (unless her will says otherwise).
Since the gift to you (assuming she didn't make any other gifts) is over £325k then you are primarily responsible for paying the IHT.
£425k - £325k = £100k
Normally, IHT is 40% but, because the gift was between four and five years ago this is reduced to 24%. So, if your mum were to pass away tomorrow then you would have to pay £24k IHT on the value of the home (£425k).
If the house was worth less than £325k when she gave it to you (and there were no other gifts) then you don't need to worry at all about IHT.
If your mum lives for another three years, then it is totally disregarded for IHT purposes, no matter how much it was worth.
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"...if she specifically wants to leave him any of her remaining estate in her will?"
The house is no longer part of her estate. What she chooses to do with the rest of her estate is entirely up to her. But the house now belongs to you.
What you should be aware of is that everyone has a tax free allowance of £325k when it comes to IHT. If she dies within the next three years then the value of the house is still counted as part of her estate and so any money that she leaves to her new partner will be subject to IHT if the house she gave you takes up all or most of the £325k allowance.
If she survives for another three years then she gets to use the entire £325k for the rest of her estate that she might choose to give to her partner.