I done ours using the 24 hour phone service for some time now ... they quote 5 working days for delivery, but typically it's 1-2.
I taxed our Audi by phone on Thursday and the tax disc arrived yesterday.
I would expect this too. If I may just add for the benefit of others.
In the back of beyond, I did mine online on the Saturday, and it arrived the following Friday, which was good, as it meant had it not, I would have had to driver about for 4 days with an expired one, as I was 600miles way from home over the expiry date.
I did contact them, and they informed me about the display requirements, and that they were unable to advise me what I can do. I had to either, not go, or take the risk if it did not arrive.
I would suspect that the royal mail is the issue, not the DVLA. So if you worry about such things, go to a post office and get it in your hand.
Now, I know this is controversal, but you do know it needs a new one every 6 months, or year, like your insurance, and you do know its due as a reminder comes. So I am unsure quite why people still rush at last minute to get one. They send out the reminder 4 weeks prior to it expiring.