The DVLA have CHANGED how they do it, you used to be able to go along and get all the paperwork done there and then.
Not now, they will give you a tax disc, but the mot and V5 goes off to Swansea. The mot will take a wekk to come back but the V5 can take up to 6 weeks depending how busy they are.
When buying my car a few months back it took 3 weeks, but my sister has been wating 3 weeks and no v5 yet.
You can't really buy the car until it is sorted as you will have no log book.