They quote a processing time of 10 working days, in my recent experience, it's taking 2-3 weeks.
In my recent experience it is taking significantly longer than that.
I renewed my license in December, and sent off the application (for an EASA "upgrade") on the 22nd December.
I received an email from the CAA on the 9th January (fair enough, it was xmas) saying that we had used an out of date version of the Course Completion Certificate. Myself and my instructor emailed them an updated version on the 10th January.
On the following Monday I rang them to ask if they had received it (knowing full well that they had because I received an automated email response from them) to be told very cryptically that if it had been received it would be logged, and dealt with in due course. I would know if it hadn't been received when I don't hear anything back from them.
I then rang a couple of weeks later to see what was happening, to be told that all was progressing perfectly well and the license would be sent out on the 28th Jan and I would receive it some time after that.
Last week I finally lost my patience and rang again, to be told firstly that all was OK, and then in a second conversation that I need to resubmit my Course Completion Certificate.
I did that Thursday evening, and rang them again Friday morning just gone to be told that it would be put in the post later that day. (Which implies to me that they hadn't actually lost my CCC at all - at the time I rang they hadn't even opened the email so couldn't have dealt with it.) They agreed to Fedex it to me at their expense because of the time it has taken, which is four weeks since the updated Course Completion Certificate was sent to them. How long it would have taken had I not chased them heaven only knows. (I guess that if they hadn't lost the CCC then it might actually have been the same time, but who really knows!)
I did ask them to email me the tracking number so that I can keep an eye on progress. It seems it's currently out for delivery...here's hoping.
As it happens I have some sympathy for the licensing department. They even suggested I email the head of licensing to complain, which is what I did. It seems to me they have issues, and know it, and are trying to effect change by making sure that the management are aware of what they have to deal with.