God this is gonna annoy some people..
I have *Never* had such amazing customer service / turnaround with a government agency as GOOD as the DVLA!
I think my replacement license was ordered on the monday and it was on my doorstep wednesday morning.
V5s can take a bit longer, but even those I've usually had within a week of the seller / garage posting their copy off.
I went to the DVLA office in Northampton to change one of my cars details and the efficiency was amazing. It was quite busy in there and all the desks were open. They even had another guy going round using the seats as a make shift desk to get people seen to quicker.