Assuming the property has never been your partner's mother's main home, if she sells her share of the property to you, she may have to pay CGT. The gain will be calculated by taking the price you pay for her share and deducting from it the market value of that share on the date she inherited it. If that occurs in the current tax year, the first £11,100 of her gain will be tax free due to the annual CGT exemption every individual is entitled to.
If the property has been your partner's mother's main home at some point during her ownership of it, then she will be entitled to some main residence relief and possibly letting relief. Take a look at HS283
for more information on that.
There are two rates of CGT, 18% and 28%. The rate or combination of rates that your partner's mother will pay will be dependent on the level of her income in the tax year she disposes of her share. Take a look here
for information on how to work out the rate of CGT that will be payable.
You will need to try to get a reliable valuation, possibly more than one, of the property from a good estate agent or land agent as at the date the property was inherited. HMRC uses the District Valuer to check valuations in capital gains calculations sent to them.
I hope this helps but let me know if you have any further questions.