Inheritence tax is only payable if the estate is worth above about £340k and only if the person giving the money dies within 7 years.
The tax would be payable out the estate, so they would have to pay, not you.
Spending some of the money on home improvements sounds like a good idea.
Whether it is worth paying any off the mortgage depends on the interest rate compared with what you can get in savings.
If your mortgage interest rate is below about 2%, you can get fixed rate savings above this level, or easy access at 1.5% so it might be worth saving the money instead of rushing to pay off your mortgage, although this also depends on what savings you already have, as there is the tax free interest allowance to consider of £1000 pa if you are 20% taxpayers or £500 for 40% taxpayers.
If the tax and mortgage interest situation makes it worth paying off some of your mortgage, you could do some this year and some next year, but check what your mortgage company's definition of a year is, is it calendar year, or within the mortgage year, which will be from the anniversary of when you took it out.
But unless you already have a decent savings pot (say £10-20k) I'd keep some back in savings anyway, but it also goes without saying that if you have any other debt, you should pay this off first, unless it's on a 0% credit card and you can easily afford the repayments.