Totally with you here, @Rosebel. Childcare is a nightmare in this country, even with an office job it was a problem for me. In my case I struggled to commute and work in the time that the childminder or school clubs operated (8-6). I worked 9-5 but the commute was almost an hour! I have no idea how people manage it with shift work.
I assume you've already looked for alternative childcare solutions? I think you'll have to speak to both the nursery and your employer and see where they do have movement. Like other posters have mentioned, nurseries usually have a 4 week notice period but if you are worried about cover, that can still be a problem.
Given COVID, and the prospect of higher future unemployment the nursery might be more willing to change what they do in return for more security on their side. see if you can offer "2 months notice to terminate the contract if i can change my hours with 4 weeks notice". Or something similar!
Conversely speak to the employer and see if you can negotiate on that side. Can you offer working Saturdays for less flexibility during the week? Or see if they can stick to 3 out of 4 days. your manager has a choice - if you can't do it you can't do it and it's in his interest to accommodate if you can.
Another suggestion is what I did, I approached one of the workers from my daughters after school club direct and asked her for babysitting etc. This meant on days I knew vwas going to be late, I could call her up and she'd take her to mine! She was already background checked & qualified because of the job and my daughter had a good relationship with her :)
You're not being unreasonable, childcare is a nightmare in this country and employers, schools etc just aren't practically set up to facilitate it.