My last numberplate was done online and took a week. The car now does not have to be taxed and insured to transfer. I don't think my car had an MOT at the time, I seem to remember (but please check) that you don't need an MOT so long as the car has had one in the last 5 yeears, this is simply to stop someone who has come across a wreck transferring the plate.
If you have a car that has been written off you can do the transfer or retention on line and it is immediate so it can be done before surrendering the V5 to the insurance company.
The DVLA is becoming much more user-friendly.