I would go ahead and do it. You're right, the chances of being called straight away are slim, but at least you're getting CV's out, and more importantly maintaining your morale by doing something. You should have the book in a week or two barring any complications. If you are lucky enough to be called I'm sure an explanation about how you are so keen to work for xyz that you just had to apply asap, will cover a few days. You know what your licence number will be anyway for any paperwork.