You both sound sleep deprived and it’s so hard. Lack of sleep raises stress hormones (cortisol) which can then make it harder to get good sleep in the small windows you do get. It sounds like their is a lot out of your hands, like her dad and your mum not helping, so I would suggest focus on what you can do, try deep slow breathing through the day and night, relaxing music, reaching out for support, getting outdoors in nature, the more your stress levels come down the better chance you have of getting quality sleep in the windows available and your relaxation may be a bit contagious to her too.
I will post a link to national sleep helpline who open Monday 9-11am 7-9pm, I rang once and found them very helpful. I also suggest contacting your health visitor for support.
If you’re sleeping lightly half an hour will feel like you have only blinked not slept, but it will all help, try and nap at every opportunity until you feel better, let housework etc go when the sleep is this bad.
One thing I used to do with my poor sleeper (undiagnosed allergies) was put a travel cot mattress on the floor by her cot so I could sleep there and pop a hand through the bars to comfort if needed while barely waking up myself.
There are lots of little pieces to try like bath before bed, nighttime associations like red nightlight and maybe relaxing music or white noise that she learns to associate with sleep time.
Is her daytime nap better/longer in a pushchair? Worth a try, then as soon as she drops off wheel her home and kip on the sofa. If she sleeps much better in the pushchair it’s possible there may be silent reflux which can be medicated or may resolve with diet changes. More daytime sleep may help her sleep better at night if she’s overtired.
Also a bit of hope, this is not forever. They change so quickly and it can all shift as they grow and as you find things that work for you. Hope you both get some Zs soon.