This is a neat idea! I'd thought of something similar before I posted this thread. My idea actually features Jonathan Archer's Enterprise and depicts a universe in which the Vulcans never joined the Federation but instead allied themselves with the Romulans, precipitating a conflict between the newly-formed Federation and the Romulan Emprie that lasted several decades, eventually reaching Kirk's time.
In this alternate reality, Kirk is still captain of the Enterprise, but his first officer is Clark Terrell (I like the idea of the two men meeting in this timeline) rather than Spock. McCoy rounds out the "Big 3" of this universe. The rest of the cast is the same as the prime universe (Scotty, Sulu, Chekov, and Uhura).
There would be no Guinan in this setting, so there would have to be another way for the characters to realize that the appearance of Archer's ship may have altered history (perhaps the Guardian of Forever could somehow be used as a referee of sorts, pointing out alterations in the timeline as Guinan did for Picard).
--Sran