Yes, DP and I work full time.
In the toddler years, we lived close to work - and luckily DP was flexible enough that if nursery called, he could work from home. More often than not, he would pick up, then wait for me to come home and go back to his office up the road to finish off his tasks for the day. And sleep deprivation does make it far harder to cope with. It was an exhausting and sometimes comical juggle at times and probably that's the stage you are at right now - it won't be forever, when your youngest is in school it will ease a bit. But for now you are in the worst of it.
During school years we moved to the village DP is from, so we had inlaws that could help but I didn't know anyone, and in order to get to know people around here, you can either get involved in the school, or in the local football team. Since I know fuck all about football, I joined the PTA, which meant I got to know some lovely parents, got to know the teachers and principal very well and networked with other committes in the village so meeting others who wouldn't have a school connection.
I also threw a whole-class party the first year of school in a soft play to get to know other parents, and that worked nicely - DS got reciprocal invites so that was more networking for me. His two best friends in school, their parents are lovely, and I know them and a few neighbouring parents well enough to feel able to ask them to pick up DS in an emergency and hang onto him for a few hours. (Did that once so I don't take advantage!) I also do a hobby one night a week for me. I started it on maternity leave and while some days I might have been too knackered to go, I did anyway because I needed a small block of time during the week just for me with nobody yelling for mom.
So I settled in, and it was nice to make friends for myself as well. I recently took up yoga on Sunday mornings so meeting more different people.
But not every neighbourhood might be like that. We are in rural Ireland so people tend to want to get to know you and welcome you to their village.
With the pandemic, we both WFH and that was just so idyllic for our family. (and saved us on childcare!) DP stayed WFH, and while I'm back in the office FT and commuting again, I now have the technology to log in from home in an emergency. If DP can stay WFH until DS is old enough to let himself in and forage for himself without burning the house down then we'll be very lucky indeed.