If the baby is only a month old, the dad could still be on paternity leave. A lot of places offer up to 6 weeks now. I had a baby last year and DH got 4 weeks full paid paternity leave and then was able to take reduced pay paternity leave for the final 2 or he could just do a couple of days a week and paid for those or shorter hours it was up to him.
If she's had a C-section or a physically traumatic birth (needing lots of stitches/ repair) she is probably struggling or not able to drive to do the nursery run. My friend had a very physically traumatic birth (3rd degree tear and lots of internal damage) and couldn't sit or drive for long periods of time for nearly 12 weeks. Even though she had had a vaginal birth (any everyone medical claims vaginal births are easier to recover from) she was in pain and struggling for months after with lots of the housework whilst everything healed. If she's struggling with post natal depression or anxiety then again she needs support with this.
It's not unusual at all for the dad to step up in regards to giving mum a quick hand with the children in the day (so she can go to the toilet/ wash/ dress/ have a drink or just 10 minutes) or to take on the cooking and more physically demanding housework jobs . After my EMCS and my planned section, I couldn't hoover, hang washing out, empty and fill the washing machine or dishwasher ( repeated bending) or stand and cook for more than half an hour at a time for a couple of months - DH had to do all those jobs when my mum wasn't with me. My own mum and dad came and lived with me for 3 weeks when I had youngest DD. I needed their help for those first 3 weeks and then they came every other day for a week. Then every 2 days until I was nearly 6 weeks pp.
I also needed my mum and DH to do a lot of the dressing/ feeding/ bathing of eldest DD and she was 4. A 2 year old and a baby is demanding. She will be tired from the pregnancy, birth and then the whole getting used to having a newborn.
They are lucky they can throw money at the problem. A cleaner, meals and a nanny will take a load off. You don't need to help if you feel it is too much.
They will be better when the nanny starts, they need to get the nanny to do the nursery drop off and then help with the baby. The nanny will also be on hand to do a lot of the childcare for mum to get washed/ dressed/ pop to the loo ect so your son won't be disturbed as often.