I found this stage excruciating (baby now coming up for 1). Like you, I was doing the baby and older child on my own because H was at work (in my case, until much later). High needs baby who hated being put down, screaming constantly and cluster feeding.
Luckily my DC1 is a bit older (5). I'd leave the baby to scream for 10 minutes, make older child some beige oven food, plonk it down in front of older child in front of the TV, bathe baby (I found this helped DC2 to sleep), feed and rock baby to sleep and then go down and encourage older child to eat their (now stone cold) dinner. Then try and do the school reading, bedtime stories etc. before the baby woke up and started screaming again. Often my older one didn't get to bed until 9/10pm on school nights and I learnt from their teacher that they'd sometimes nod off during carpet time at school.
Mornings and having to get out in time for the school run were an absolute shitshow.
I got through it by essentially neglecting my older child. Let's face it, a screaming baby is always going to win out over a fairly independent, chilled 5yo. DC1 watched hours of TV. Still has too much screen time tbh. I tried to make it up to DC1 at the weekends when H was around/family were visiting.
What have I learned from this? Firstly, that DC1 is an absolute star - has had to put up with a lot but still adores their baby sibling. Second, I'm pleased not to have had to do this with a 2/3yo. Third, if I did have to do it with a younger child, like you do, I'd get help in a second if I could afford it.