When it comes to ME3, I've already said almost everything I'd want to say about it, but this is relevant, so: I think the writers had the perfect opportunity to use the two-year time gap between ME1 and ME2 to fill in these blanks. Maybe Ashley (or Kaidan) was able to work her way through the ranks quickly because during that two year period, she won some key victories in a couple of engagements with Cerberus. This would not only explain her jump through the ranks, but also establish her credibility as a capable soldier apart from Shepard and give her a reason to be resentful and distrustful of Cerberus that even your alien squadmates would have: More than them, she would have seen up close and personal what they're capable of. Anyways, that seems like it would have been the best way to fill in multiple story gaps IMO.
That would only go so far.
Killing enemies does not translate to officer material. Especially such a massive rank up over such a short time period (1 year from mid-level staff NCO to a field-grade officer. That just does not happen, unless you are a very senior civilian specialist working in a very specialized military project, as nearly happened with the Manhattan Project, where Dr. Oppenheimer was almost given a temporary direct commission to Lieutenant Colonel.)
And as a Soldier, I question from the get go her credibility as one. As well, once again, credibility as a soldier does not equal credibility as an officer.
Honestly, the only thing I can think of that got Ashley that far that fast is nepotism/cronyism; she used her association with Shepard to ride his coat-tails to get where she is.
And when Ashley implies that you're accusing her as such in one of the convo's in the hospital, I wish there was an option to confirm that you feel that way about her promotion to Spectre.
My view is that after Shepard died, Anderson/Udina gave her a pity promotion and a chance to revive her career and/or gain a useful contact within the alliance command chain. She used her association with Shepard to get herself moved forward, and, sans a few missions here and there, never really did anything to earn her rank. Same with her Spectrehood, though that's somewhat offset by Udina wanting a pawn as well as being a political consolation bone being thrown to humanity over the loss of Earth.
It's something that's happening all over science fiction too. In Halo, a universe that is traditionally very militarily accurate (they use United States military procedure and rank structure for the most part) is starting to fall into this. You have the character of Sarah Palmer (IMO, Ashley Williams' analogue character in that universe, whom I also utterly disdain) going from a junior enlisted Marine (albeit a qualified and experienced ODST) to a Spartan Commander, with notional rank over even the legendary Spartan II's and III's. Granted, this has been mollified slightly with the revelation that she isn't an actual Naval Commander, and her rank isn't actually one either, but more of a title. On the other hand, the Master Chief was offered a direct commission to a flag officer, though this is much more justified given who and what the Chief is, with more than 40 years of military experience and leadership under his belt, and having reached a point where even 5 star flag officers are deferring to his judgement and word prior to his offer of commissioning.
Point is, you don't go from a nobody with a black-listed name and no experience to a field-grade officer who is made into a spectre in 3 years unless you have somebody seriously pulling strings for you.