The transfer should be pretty easy. Fill out a AP1 form which you can download from the .gov website. You’ll also all need to fill out an ID1 form to prove identity. You can then get a solicitor to verify the ID1 form for a fairly small fee a post the lot off to the land registry or you can take it to the land registry in person and only pay the transfer fee (£40). They have regional offices but you will need to book an appointment and all go with passports. Or you can get a solicitor to do the lot if you aren’t comfortable with that.
Not so sure about the tax. I think even if they gift it they will be liable for capital gains tax on the market value. They will both have a cg allowance so it might be under the threshold. Probably too late to split it across two tax years now. If they gift it there is also a potential inheritance tax if they die within 7 years. You should be under the stamp duty threshold.