Just assume the Transwarp beaming thing is a 24c technology.
The way I figure it is this: Scotty comes up with the idea (in primeverse) around the same time he does in nuVerse, but doesn't have the tools or know-how to really make it work. And it becomes one of those projects people tinker on throughout their careers. He uses Spock to bounce ideas off from time to time since he's the one mostly likely to have some insight. He eventually brings it with him on the Jenolan
as something to do in retirement and mentions it Picard in passing. Geordi, Data, and O'Brien hear about it and ask Scotty to show them. He does, and then it becomes their life-long project.
Eventually one of them cracks the nut. Spock hears about it and calls 'em up to ask because he had an invested interest.
Then, of course, he ends up taking it back in time with him. nuScotty is able to reverse-engineer (The Chiefs?) wizardry because it was still so close to his own.
Khan can do the same because he's uber smart.
Yeah. That's really convoluted and probably not worth the effort. It's best to just say to yourself "It's magic!" and move on.