Its dead easy, I did mine.
Go to the CAA and ask for the one you want and make sure its available.. You then reserve the reg. Pay the fee to the CAA and then technically you are grounded because your LAA papers are invalid, but if you speak to the LAA office it is possible to get everything reissued within a week ( don't forget the new fireproof plate). There is a small charge for this.
All should take less than a week , speak to the LAA before you do it and they will advise you how long the re issue will take ( ie if the person that deals with it is on holiday for instance , you don't want the papers sitting on their desk for two weeks).
Pre order your nice shiny new vynal letters from any reputable signwriter and the world's you oyster! or whatever reg you want!
Edited for cost ...All in about £400 inc new vynal numbers