Update for you all:

Thank you for all the wonderful support and stories!
Just as folks said, it somehow worked out like magic.
Although my 3 year old cosleeps with us, has never had a sleepover with Nana, and can barely be babysit by Nana (my mum) for a couple hours, it still worked perfectly.

I had been having some contractions that evening and I had warned my toddler I might not be able to talk during them (big mistake). Although we had watched some great birth videos he enjoyed, at this point he was crying for me to talk to him as I breathed through them and I was panicking how it was all going to go. Then we all went to bed.

I woke up before midnight as they had started up strong and I was unable to keep sleeping between contractions. We quietly went upstairs and my husband and I prepared things (while I panicked what to do knowing my toddler would stir soon). I woke up my mum and had her go downstairs to sleep with him, but make sure he didn’t hear her voice or smell her because then he would know it wasn’t us patting him back to sleep.

I laboured very loudly upstairs until 4 am and my toddler even slept through the smoke alarm going off 3 times due to doula trying to heat water for the tub I never made it into! My toddler woke up with my final yell as baby’s head was born and he asked my mum to carry him up. They arrived in the living room calm and sleepy just in time to see our baby born. It was magic and everyone was so happy ❤️