When I last changed my car I removed the personal plate (physically and with the DVLA) when I was parked at the dealership where I was part exchanging the car. The DVLA sent me the V5C with the changed reg for my old car a few days later, I just posted this to the dealership. I think it was the dealership who suggested doing this.
To put the plate onto the new car I think I had to wait for the V5C to arrive, but that didn’t take too long.
The other way of doing it is to remove the plate now, and then give the documents to the dealership to put on the car before you collect. I can’t remember why but the dealership I bought my new car from weren’t keen in this option – I think it is just more hassle for them.