I'm sorry you're in such a predicament. Is your DS not going to nursery much (as others have commented on) because they are strict on their policies and allowing unwell children to be in?
I work in a nursery, and we will only send children home/say they need to be kept at home, if they are vomiting/have diarrhoea (in which case they can return 48 hours after the last bout, if they've got covid, conjunctivitis, hand foot and mouth, etc. There are other, specific, illnesses that mean a child must stay off nursery, other than those I've mentioned. We do allow children with colds in if they do not have a temperature, which we check regularly if they do have a cold. Here's a link to NHS site, hope it helps.
https://www.nhs.uk/live-well/is-my-child-too-ill-for-school/
I do agree with other pp that if your son if unwell frequently, medical advice/attention should be sought. And also as someone mentioned, look into parental leave.
You've mentioned you can work remotely; depending on how your DS is, I don't think most companies would have a problem with it, as long as you're able to do your work/get projects etc completed and if on work calls not have to frequently step out. Maybe see if your company have a flexible working policy - you may be able to alter your hours. It would mean cutting your pay slightly though.
Best of luck, and I hope your son feels better soon x