Typical waiting time for a PPL of 15 working days/3 weeks is also what I was told by the CAA, and the CFI at my flying school.
In my case my examiner didn't submit a test notification to the CAA prior to the skills test as per new regulations, and the CAA kicked up a fuss about it. I actually ended up waiting 98 days for my license to arrive from the day the CAA got my completed application, and ended up missing out on a course for my ATPL theory (PPL required to undertake certain aspects).
I would therefore suggest you double check with your examiner that this has been done, otherwise you are in for a world of pain with the CAA.
I would also double check you are actually allowed to fly PIC without passengers before getting your license. When I passed my skills test, and started hunting around for a place to hour build at, at least 3 schools/clubs insisted that I would not be permitted to fly without an issued license. When I phoned the CAA, the CS rep I spoke to was very adamant that I would not be able to fly without a license (passengers or not) and that I should not log any hours. There were a few schools (including the one where I did my PPL) that say it is OK, and I know plenty of people do continue flying before license issue, but I figured I should probably just play it safe and wait. I will now need a checkout, but it's better than possibly causing grief for myself with the CAA later on.