I passed my test at the start of December, applied by post and had the licence through the post by mid January.
They were closed for a week over Christmas and were very busy at the time so I think under normal circumstances a few weeks is what it takes.
As others have said though, make very sure you have all the relevant paperwork filled in, no missed signatures, everything in order or it's sent back, at your expense, and you start from square one again.