If she's looking "very pregnant", then you're almost certainly very close to the birth. Like, next day or two.
If you are comfortable allowing the birth to take place on your balcony (which would be a helpful thing to do, as having them more in the open exposes them to getting picked off by crows and other predators), then you might want to put a cardboard box outside ASAP with a couple of blankets, or perhaps some straw-type bedding (which will stay "fresher" longer than a blanket, which will get matted pretty quickly and can't provide warmth when it is matted down at the bottom of the box). You can get straw-type bedding at a pet store-- it's sold for hamsters and rabbits.
If you can get the rescue group in there beforehand, that would be best, of course....
Instead of the no-kill shelter, do you have any rescue groups that might be able to help? They would try to rehome the babies, and the mother too if she appears able to be domesticated.
You might want to start a separate thread asking for help in locating a rescue group, putting your location prominently in the thread title. Alternately, maybe the no-kill shelter has a list and can give you some telephone numbers?