Plates on your car should always match the tax disc - fit them when you get the new disc from DVLA. They turn transfers round pretty quick, I'd say less than a week.
But you will have to wait longer for the new V5 to come through. Can be up to 4 weeks.
My last transfer, last week, took 5 days, including the weekend. This was Bournemouth office. They've been fairly consistent over the last couple of years.
