Both places I've lived here have taken just around a week. It usually breaks down to 1 day for viewing and application (and usually this comes a few days after you call... I've never experienced a same-day viewing), 2 days for reference checks, 1 day to tell you it's yours, and another day to get the lease signed and get the keys.
When we had to leave our last place, we started looking on the afternoon of the 25th and were moving and getting keys on the 1st. Of course, we also had local references and a full year at our previous address, and a bit of credit between us.
That was a year ago, though. I know in some markets places aren't moving, so everyone is really eager to get a tenant in fast. I was speaking to an estate agent on Friday, and when I said we would want to move around the 1st, she actually asked if we'd be willing to move in sooner.