There is definitely no counter service for same day issue of an initial licence.
You can still make use of the counter service in the sense that you turn up there with all your paperwork and they check through it there and then and take the payment. Apparently this speeds things up by a few days.
I dropped my stuff at the counter on the 9th and my licence was delivered on the 28th - so just under three weeks.
Either way, you'll be absolutely fine for Aero Expo as it's not until the middle of June.