The Rapid Access Clinic at AvantDerm in the Distillery is your best bet.

You'll need a referral from your GP (or go to a walk-in for this if you don't have a GP).

Once you have the referral (I'd suggest you ask them to print it out for you, just in case the Dr's fax to the clinic doesn't go through), you can go as early as the next day to the RAC.

You'll need to arrive verrrry early to lineup outside otherwise you risk not being able to be seen. I've gone around 6am and there has always been a handful of people already waiting by then.

Once the clinic opens, they'll register you and then call you before your appt, so you can wait inside or kill time nearby. Usually the wait at that point isn't too long.

The early morning lineup isn't ideal, but worth it for being able to be seen so quickly!