Get a valuation from a RICS surveyor or 3 agents and agree price. For ease I will assume the property is worth £300k. Work out increase from probate value. This is amount that is potentially liable for CGT. E.g. probate £150 k so increase is £150k
This where I struggle to advise so definitely check it out, I think DS1 will be liable for 1/3rd of increase e.g.£50k, and has a CGT tax allowance of £12,300 but may also be able to also use any unused CGT tax allowance from previous years. www.gov.uk/capital-gains-tax/allowances
This also suggests disposing an asset to family members may impact so definitely get legal advice. Note:CGT is paid at 28% and would be paid by DS1.
Discuss and agree with DS2 is do they want to also buy out, so you each pay 1/6th (£50k) and own it 50:50 or will you buy DS1s share and so it’s owned 33:67?
Note: make sure both DC know and understand DS2 could end up with more in the long term but will have to put cash in now. This could be an issue when DS2 realises their inheritance.
Having decided this you need to fund the buy out, cash from savings or a buy to let mortgage. Either way DS2 has to agree to a mortgage but if you go down the 50:50 route DS2 would be responsible for paying 50% of the mortgage.
Note: DS2 needs to understand his share of rent goes up but it would be used to pay mortgage rather than going in his savings.
If you opt to own it 33:67 then you need a deed of trust, or similar, to reflect your respective shares.
Essentially then it’s like buying a house there will be a transfer deed which transfers ownership and the money is paid to DS2. You may, particularly if you go down the mortgage route, have to do searches, have a survey etc. these are normally paid for by the “purchasers”.
I would also consider inheritance planning for you and DC in terms of wills etc.
Finally, consider, even though this is family, if both DS (as well as you) should have independent legal advice/financial to make sure they fully understand the implications for them. Yes there is a £ cost, but in terms of family relationships it may be money well spent.