I think one of the strengths of "Yesterday's Enterprise" is just how well sketched the alternative universe seems to be, without it actually going into too great detail. It just does it all with broad strokes. But that's what makes it so compelling, because it seems real enough
to convince us that it was a valid alternative path to the 24th century we 'know'.
To the broader question posed by the OP, I can't imagine that the Federation changed so
radically all at once. I imagine that following the Narendra III incident, tensions escalated out of control, but people like Ambassador Spock maybe still tried to negotiate and maintain the Khitomer accords. I can only imagine that the Klingons made some major attack that caught the Federation with their pants down, and therefore the Feds had trouble regaining ground.
One thing that needs to be remembered (but often isn't) in these discussions is that even canonically, in our own 'real' TNG universe, the episode "Aquiel" stated that the last Klingon raid on Federation territory was in 2362, only two years before the Enterprise-D was launched. That's not an alternative universe, that's a solid fact that happened in the 'real' version of Star Trek history.
So, it is apparent from this incident, and from events in DS9, that the Federation/Klingon treaty remains fragile even in the supposedly peaceful 'prime' universe. All I can imagine is that the tipping point that turned everything into a war zone was that one straw incident that didn't happen in 'our' Star Trek universe. "Yesterday's Enterprise" itself evidently implies it was as simple as the Enterprise-C not being where it was supposed to be, but I reckon there was probably in 'reality' more than that. It was an escalation of events.