Up here the service hasn't been quite as far privatised as in England, and we have a surgery 500 yards down the road, so can generally get in either by waiting on the day or appointment within 24 hours. (I won't link this to the Referendum!)
But a couple of years ago, I got what turned out to be shingles, so while passing a pharmacy in the next village mid morning, went in and asked if they could have a look at the red rashy bit. Pharmacist said instantly "that's shingles, who's your doctor?". Gave the name, pharmacist rang the practice and got the reply "if you can be here in 20 minutes, we'll see you". It was an 18 minute drive to the surgery, but made it and GP prescribed some pills, but they hadn't got them in stock in that surgery. They rang one of their other surgeries, which had them and "if you can be here in 10 minutes, we will still be open" (they are closed over lunch). Result.
Moral seems to be that you may get better access by being referred via a local pharmacy.