I find myself in this exact situation, although I have to finish painting four units in the next nine days, and I have to summit my army list for the tournament I talked out in my last post today, well yesterday now as I'm a bit late posting this up.
These are some small changes and learnings I have made to ensure I have a chance of hitting my deadline. This are not tips on how to paint to your best quality, but how to speedup getting a unit painted, so some of these tips are not for those looking put out their best work.
If you're painting in the evenings like I am you need a good light source. Ideally you should use daylight bulbs, as normal bulbs can give a yellow glow meaning that when you check out your masterpiece in normal daylight they don't look as good as you imagined when you finished them the night before.
Also it is best to have two lamps from different directions to avoid shadows. Be mindful to keep the lamps far enough away, because otherwise the heat from the lamps will dry out your paints much quicker meaning you have to get more from the paint pot more often.
Painting the Visible
If you like me you've probably spent hours over all the models you've painted trying to squeeze paint brushes through small gaps to paint a part of a model you can only see if you look at the model from an obscure angle that no-one will ever see. You must stop yourself doing this. Concentrate on the areas people will see, face, chest, shoulders etc. It's much better to finish more models than be happy that every small detail is done on a fewer amount. This is particularly important if you are painting an army and have a deadline to meet.
Painting those hard to reach or see areas will add nothing to the overall appearance of the model, but will seriously increase your painting time, especially when you consider the touch-up required on other areas you might accidentally touch.
Standard is your Friend
Try to use standard colours rather than shades you mix up yourself. Having to mix up just the right shade over and over again will cut into painting time.
If you do want to use your own special colour. Write down the colours and proportions used, and create the colour in bulk using a mixing pot. Not only will you always have exactly the right shade every time, but you will save loads of mixing time.
Let Friends / Family Know
If you have a tight deadline like I have, let you friends and family know that you are busy each evening that week. This will avoid (or at least lower the risk that) your other half will request a romantic evening on the coach watching the latest RomCom, or friends organising the party of the century.
Alternatively organise something for after you finish you painting deadline. It will soften the blow that you won't be available while your painting, and it will give you something to look forward to while you paint.
Make a schedule to keep you on target. This will give you a target to aim for each planning session. Be careful to leave gaps in the schedule which you can use to catch-up if you fall behind, because things happen in real life which come above painting which have to be done (in my case I have to prepare for an job interview).
Lastly if you are really up against it, you can leave some techniques to later, eg. you could come back to your models for the last highlighting sections etc. Once you've got a portion of your army done it will inspire you to set yourself another target to get more of the army done and/or go back and add a layer of detail to the models.
My space wolves are progressing slowly, but the remaining weekend I have left for catch-up and I only planned basing for next week means I have a good chance of getting them to a level I can use them at the Magnificent Seven tournament:
Please share your ideas on how to speed up painting.