There is a growing trend where employers are making it so managers cannot provide references, they have to forward all questions to HR, which will only confirm your dates of employment title, and salary.

All of my jobs I got while still employed so I couldn't use my direct supervisor as a reference. I used either a peer or someone who was as high up as my supervisor who i jaf worked with closely who I trusted. It worked out pretty well. So it is possible in fact, I've done it three times in a row.

So find a peer or another supervisor who will vouch for you. If worse comes to worse, explain that they won't give you a reference because you had to leave with no notice due to a severe emergency family illness. I think most reasonable people understand those circumstances and that they are out of your control. I think most reasonable people, at this point, will recognize that two weeks notice is ridiculous now, considering employers can terminate you at any time without severance.