Yes, they can usually put your private plate on retention and assign a new one on the same day. You might need to leave it with them for a couple of days if they are busy, but that would be the longest I would have thought. The longest I have ever had to wait was 2 days, the quickest about 2 hours (you can call back later to pick it up usually, you don't have to sit in the waiting room!).
As I said before, if the car has never had a standard (non-cherished) plate, they can't do it there and then, and in that situation it can take about a fortnight.
I would say that as long as your car has had a standard plate on it at some point in its history, the only issue you will have to worry about is getting the new keeper's details on the V5, as you will have to wait until the V5 with the new number is sent to you, then fill in the change of owner stuff, then send it back to Swansea again so that the new owner gets the V5....
I think that I have made that sound more complicated than it is