How to improve writing skills when writing a manuscript for the first time?
Concrete advice on where and how to begin, especially given that I
  don't have that much time


I work in biomedical science so my advices may not fully apply to you, feel free to take whatever that is useful.

Don't write too much: Most students I work with who are trying to turn their paper into a journal article tend to write too much. It's usually described as a painful process because it would feel like chopping off a lot of work from their thesis (which was done with much blood and sweat.) But the trimming and refinement need to happen because thesis format is not suitable for journal.

Concretely, these newbie articles tend to


Have an introduction that is overly long
Contain abundance of details in the methods section, documenting the conception of the idea to every little details, with meticulous follow up of citations
Justify every single decision, some of which trivial
Cover too many hypotheses or fail to boil down the structure to bite-size, thinking everything in the chapter has to go into the paper
Elaborate lengthily in Discussion, failing to highlight the most important aspect


But also don't write too little: And yet, most students tend to overlook these components:


The paper goes into technical way too abruptly, doesn't account for the fact that some non-specialist may be reading this. And this is really an art, you'll need to see your target journal and get a sense of who are the readers, and then decide how broad to start with
Fail to explain the "so-whats" in the Discussion
Did not address the strengths and the weaknesses of the study. Or quoting weaknesses that actually can be alleviated but were not. For instance, I have a lot of students stating weaknesses like "this is a cross-sectional study so causal inference is not feasible," which I tend to agree; but some may say "in this analysis we did not adjust for income levels" while the income level variable is actually in the data set. In those case I'd request a re-analysis. In short, if you can have done something about it, don't list it as weakness.


General style mismatches: Sometimes the articles have nothing really wrong about it but the style just make it clunky as a journal article. For instance:


Overly sensationalizing or flooded with emotional adjectives. E.g. "the obesity epidemic is indeed a human tragedy," "our results are surprisingly encouraging."
Mixing up facts and arguments, ending up with a lot of statements that are not backed up or statements that are accompanied with a tsunami of facts without a trace of synthesis.
The objectives and the conclusion do not match. And this is very common in theses because we set off with a plan and things may fail to work in the middle and we have changed direction. In journal article, the objective statement and the conclusion need to resonate so that they read nearly like Q&amp;A.
Trying too hard to make the whole paper self-contained and complete. Common signs are including too many details that can be included in the form of citations.
Calling the same concepts or things with many different names. In English composition we are often encourage to diversify our word selection but this is a big no-no in scientific writing. If the variable is "occupational stress," call it so throughout. Do not use variants such as work-induced stress or even worse: work-related mental problems or job-site related depression, etc.
Drafting, writing, and editing at the same time. Most of the students lost steam because they want to churn out perfect sentence from the get go. This causes the process extremely slow and painful. It's easier to do a brain dump and then come back to edit the work. On a good day you may save 50-60% of the dump, on a bad day maybe 10%, but I found it worth it because I can be more focused in either laying out ideas or perfecting the wording, separately.




So, how to start? Before writing, read a bit. Download the "guideline to the authors" from the target journal and use a highlighter to indicate all stylistic requirements. Follow them to the t.

Then, download about 3-5 articles related to your fields from that journal, and read for word-uses and syntax. Generally, when reading each paragraph, ask yourself "Why did the authors put this information here?"

Now, when it comes to writing/drafting. I find it easiest to do in this sequence:


Identify 1-3 main analysis outcomes (table, graph, regression model, etc.) Paste them on a wall or a board and they will be the staple of this article.
For each of these output, write about 2-4 main points or interpretations people should take away with. Imagine you're at a conference manning a poster. What would you tell the audience to focus on in each table or each graph? If one of them has overwhelming amount of talking points, you are trying to pack too much. Consider breaking the analysis output down, and trim, trim, trim. In this stage, do not throw away unused outputs, put them aside in a folder or a box in case you have a second thought.
For each of these output, write out the methods. Remember to only contain enough information to allow readers to replicate, not too much so that the readers have to endure the whole story. For instance, reason for picking a certain concentration or threshold can be explained with a citation rather than writing it out.
Now combine your selected output into the Results section. Write it in the driest style you can think of. No interpretation, grievance, celebration, defense whatsoever, just plain, hard, cold data.
Break Discussion into segments A, B, C, D, and E. In A, use one paragraph to summarize what you found. In B, incorporate the talking points you generated for each of the output, so that the first paragraph makes sense. In C, compare and contrast your work with other literature and discuss the potential differences. In D, delineate the strengths and weaknesses of your study. For each weakness, discuss what you had tried to make it less a matter, or what else could have been done in future work. Finally, in part E, go slightly wider and wilder: elaborate what these results mean to your target audience and their field. You can quickly see that this section can get overwhelming if you have too many objectives. For starters, one main objective with a sub objective will be a good challenge.
Write the Introduction. Focus on some "what's known" to bring your audience to the same page so that they can understand your work. Then, discuss what is not known or what challenges the field is facing. After that, explain how your work may help, and lastly, lay out your objectives clearly.
Cut and paste the objective at the end of the paper (or use split screen so that you can see the objective section), and start writing the Conclusion. Introduce no more new information there, simply answer the objectives point by point. Make sure the objective part and the conclusion part resonate. When you feel it's good enough, you can then take out the reference objective you just pasted there (or close the split screen function.)
Go back to write the abstract.




Other nitty-gritty

Don't feel bad that people are not happy with what you write. A lot of the time they do not even concretely know what's wrong with the piece. Ask for actionable suggestions and then thanks for their comments; take whatever applies to you.

Your supervisor may not appreciate your writing quality, but will definitely appreciate your attention to details. So, make his/her life easier by following the journal's guideline tightly when it comes to formatting.

Check if all the figures and tables are correctly indexed.

Check if all the pronouns connect well and leave no room for misunderstanding.

Use subtitle to introduce another layer of structure if you feel too overwhelmed.

It's true that some papers can take years to write. But you have been working on it for more than 3 months and it is more than probable to write a decent article in 3 months. Some other users may say a paper can take years, but remember the duration is also a factor of i) how much time the person is putting in every day, ii) how much the work is being held up by co-authors, and iii) personal writing style. I myself have papers that were done in the span of 2 months to 6 years; there is pretty much no rule when it comes to time.