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I'm a PhD student and I need to (almost desperately) squeeze in another paper for my dissertation in the next 6 months. Is it acceptable to explain my situation to the editor and ask to set a tighter deadline to the reviewers?

My field is health sciences and biostatistics.

You can ask for just about anything, but unless you already have some connection to the editor you aren't likely to have much happen out of the ordinary. In particular, they are unlikely to move you ahead of another author and you would probably object if the situation were reversed.

The editor has little control over what his/her reviewers do or their schedules. If he/she tries to press them, they might just turn the paper back and you get in line again. The editor is unlikely to move on a paper without reviews. The time it takes is the time it takes.

Note that in some ways the reviewer "stable" is more important to the editor than the current bunch of authors since there is a long term relationship between editors and reviewers. Since they probably aren't paid, there is little leverage.

But you can ask.

@Buffy is right that you're unlikely to be able to influence processing speed for a given journal.

One of the axes on which journals compete is their processing speed. SciRev provides a platform for comparison (e.g. here are the reviews for statistics journals, sorted in increasing order of "total handling time":
note that reviews are provided by authors, and the number of reviews per journal varies enormously, so you should be appropriately careful interpreting the results). Googling "academic journal processing time" points you to a variety of other discussions/comparisons. Sometimes journals provide this information on their web pages.

Thus you could aim for a venue with rapid turnaround. Unfortunately for you, in my experience there's a correlation between selectivity/flashiness and rapid turnaround (e.g. high-impact journals often ask for reviewers to return reviews on a very short time scale), so you might have trouble getting your paper accepted in such a publication. It may be that some of the less-traditional, more open venues such as PeerJ have relatively fast turnaround times.

You obviously know the policies of your advisor/institution better than we do, but (again in my experience, in not totally unrelated fields) it's often sufficient to have a paper submitted to a reputable journal for it to count toward a dissertation; this rule prevents the committee from having to deal with crappy manuscripts in an early stage of preparation, but insulates the students from the vagaries of the publication process. I'd definitely recommend double-checking with someone knowledgeable ...

You can ask but it's not likely the editor will say yes, unless the journal is desperate for papers.

It's not true that editors cannot speed up papers even if they don't want to impose on their reviewers. Paying more attention to a paper speeds it up - for example, instead of inviting five reviewers to start, invite ten, and the chances someone agrees and writes a fast review goes up even if one doesn't change the review deadlines. Similarly if reviewers decline, inviting new reviewers immediately will probably speed up the processing time.

However the question the editor will have to handle is, "why should I bother?" It's not fair to other authors if he spends less time there, and he can't speed up every paper, so why yours? Does he know you? Does he need you as an author? Is your paper likely to be good? Unfortunately my read is that unless the journal is desperate for papers, the editor is not likely to say yes unless he takes pity on you (maybe he experienced something similar as a PhD student?).

I think it is completely reasonable to ask.

I would go even further and look for a journal that promises fast processing (a letters journal) and submit there. But also let them know your speed need/desire.

And of course pull it if nothing is happening with it or if it gets all snarled after some period of time (couple months).

BUT. Make sure the paper is very CLEAN. No wild claims. Just the facts, ma'am! Well written. Follow the formatting instructions of the notice to authors like a nuclear tech. Make that thing EASY to fly through like a greased pig.