I've just done this in the past couple of days. The MOT is not re-issued. The MOT you had on the previous plate is valid until it is time to renew. I checked this by calling the DVLA.
The process now is immediate....I bought the plate on the phone, the supplier of the plate did everything online with the DVLA and within 30 seconds I got an email from the DVLA to state that I can start using the plate, and that was on a Saturday !
The new V5 arrived today.....within 3 days.
The most difficult part of it all was getting plates made up without the V5. I had to go back to the company that I bought the plate from and they sorted out a set of numberplates for me, as I needed to use the car. In actual fact I could have run it on the original plate till the V5 turned up but I was eager to have them fitted asap
Avoid using online companies that do not provide a phone number to have the numberplates made up.