I have no problem with the timeline as such: it's just the "threatens the galaxy" aspect that calls for a bit of explaining.
It's quite plausible that Spock is the only politician of any clout to take seriously the rantings of some mad genius who says "Hey, this perfectly well-behaving star is going to explode in a few weeks, even though nobody else can see the signs". The leadership of Romulus will simply go "This is crazy - we know
how stars work, and this one is not going supernova any time soon", and it's just that this time they are dead wrong, literally.
Plenty of time for Spock to beg people on both sides of the border to take the problem seriously; less time to work out the red matter solution once the Vulcans accept the truth; and too little time to deliver the solution before the star goes kaboom.
As for the "threatens the galaxy" aspect, it could simply be that the loss of Romulus would threaten the galaxy - with political chaos. On the other hand, Spock does seem to worry about stopping the supernova even after Romulus is already gone, as if he only had minutes in which to stop further havoc, rather than the decades that would be involved if a realistic supernova kept expanding towards neighboring stars. Perhaps this is a hyper-supernova of some sort (possibly even artificially induced), and it is exactly because of its exotic and unique nature that people refuse to believe in it until it is too late?
Star Trek makes mention of several advanced, supposedly interstellar civilizations ceasing to be when their homestar goes supernova. Probably not overnight, but still. Supposedly, then, the evacuation of a couple of Romulan bigwigs to another planet in the Star Empire, and the establishing of a new government, would merely slightly delay the inevitable: by some unknown mechanism, the conquest and dissolution of the Star Empire by its enemies would be assured.