I passed my Skills Test in April (yay!)
and on the next working day I hand-delivered my paperwork to the CAA at Gatwick (after the flying club 'chief' had checked everything was in order first).
I was quoted a lead time of three weeks and exactly three weeks to the day I received a phone call to say that I could collect my shiny new licence!
Whilst I was waiting for the licence to arrive, I asked if I could fly solo, and I was told by my club that I could, as I would be flying on my instructor's licence still.
Hope this helps.