You go to your solicitor.
Let them take up the Deeds whether or not the mortgage is paid off. Both owners will have to sign an authority.
They'll prepare the Deed/ Transfer and associated declarations for X to sign.
They'll set out your fees, stamp duty, registration fees, etc.,etc.
I'm thinking you may now have a Capital Acquisitions Tax liability- I'm no expert - you'll need to take tax advice on this
See this link
"because we want it done straight away and with no fuss of banks and paperwork etc. "
Just because you want it done straight away, that does not necessarily happen.
I love when clients "don't want any fuss" - this usually means "I'm not listening to anything you say that I don't like the sound of but I'll be real happy to blame/sue you when what you said that I wouldn't listen to actually happens. "
This really is not that difficult but you take your advices before you finalise matters- not afterwards.
Your solicitor will send the necessary paperwork to her solicitor , she signs it and Bingo!- all sorted.
It's a necessary project- not a conspiracy.
mf