If one person buys the other out of the property can they both remain on the mortgage?
Technically, if done informally, the answer is yes. Eg one person could give an agreed sum of money to the other person and they would agree between them that the sum of money represents their equity in the property and both parties agree that the person bought out would no longer pay the mortgage or have a right to ownership to the property.
However, no sane person should contemplate doing this, as doing this informally and leaving both parties on the mortgage and the property title means that the person bought out is still legally responsible for the debt, if anything goes wrong with the payments AND it would be seen as a debt against them in future, for example if they wanted to buy a property a year or two later, they would already been seen to have this debt against them and it would impact affordability checks. A second point is that even if the financial/mortgage side worked out, by remaining on the property title, legally they still own half the property and could enforce that at any time. Eg 10 years down the track, when the mortgage is paid off, legally being on the title means they own half the property and could legally enforce that ownership.
If so does that affect either party?
See above answer. There is more, but you’ll need to be more specific about what specific info you are seeking.
How easy is it to take someone off of a mortgage?
You aren’t “taking someone off the mortgage”, you are effectively ending that mortgage and applying for a new mortgage in one person’s name. At the same time, the property title also changes from two people to one. So the real questions here is really “how hard is it to apply for a mortgage?” And the answer will depend on your salary, other debts, equity in the home, valuation of the home etc.
Does the remaining person need to prove they can afford to keep paying it?
Yes, see above.
Does the remaining person need to set up a whole new mortgage?
Usually, yes.
How straight forward would any of this be?
This depends on your salary, equity in the home and house valuation. If your house is valued at £500,000 and you have £450,000 equity in it, meaning you would need a mortgage of £300,000 (ie the £50,000 left to pay and £250,000 equity to the other person), plus you have no other debts and a salary of £120,000 then the mortgage affordability calculators would have you well within range. This would also be impacted by your age. If you are young and have time on your side for a lengthier mortgage, this will help. Depending on the exact figures, it could be more or less difficult.
Anything else that may be helpful to know?
There is a lot more to know, but without specifics, it is very cumbersome to advise further and may be a lot of non-relevant information.