The stories are sketchy, but what I believe happened would have been the community hospital in Ayr, would not have been geared up for re-implantation of digits, so had to arrange for transport to a more centralized facility which does. Being that Ayr (pronounced "air") is quite a rural town, and given that his injury was non-life threatening, he would have probably have had a longer wait for an ambulance. The "bag of peas" is journalistic nonsense - he would have been assessed, medicated for pain, and had the finger chilled prior to leaving the facility.
Ayr to Glasgow is about 30 miles north east, basically on one straight road (A77) with minimal congestion.
Definitely not the ideal situation, but one similar to which I received a week or so ago. Gent fell through a wooden bridge, got hung up on his ring finger which got yanked clean off. Transported by BLS Volly crew. No pain management of course. Result - not even an attempt at re-attachment due to it being pulled off, rather than sliced off.
So, not much different than the above story.