i have changed over plates a few times
if the car is registered in your name then simply take the log book ,mot, and tax disc in to your local dvla office , they do it all there and then ,they print you out a mot, new disc,and they give you the green slip stamped with the new reg,they also give you a letter to enable you to have number plates made up..took half an hour..simples..!!!

it does however takes time if the car isnt registered and you have the full log book) to the person trying to do the change