@AngeletC, sorry just saw your post.
I used to live in Dubai, I recently went to visit and managed to get the PCC during my trip. As long as you have all your documents copies and passport pictures, the whole process takes 3 days.
The PCC is done at the Dubai Police General Headquarters near Al Mullah Plaza. Just be aware that there are a couple of Police departments in that area, so make sure you goto the one closest to Al Qiyadah Metro station, that's where they do the PCC. If you have trouble finding the place, I would suggest calling them ahead of time to get directions.
Good Luck.