I actually rather like the idea that McCoy's entry into Starfleet happened in roughly the same way and at the same time in the prime reality as it did in the altered one (apart from meeting Kirk at the same time though, as Kirk would already have graduated at that time, thanks to his father's influence). Perhaps Starfleet Academy is where McCoy met and became briefly involved with Nancy, from "The Man Trap"?
My thought in working on McCoy's backstory for A Choice of Catastrophes
was that it's the same until he enlists in Starfleet in 2255. In the Prime universe, he's posted to Republic
after enlisting (with Nancy Crater), and completes training onboard ship, while in the New universe he undergoes that training at Starfleet Academy.