At this point it's get ANYONE in to help right? A relative? An au pair? A nanny? Screw the rules.
We struggled too - I didn't get a place for my child as a key worker (even though I am one), so was teaching from home/in and out of school covering own key worker kids classes, partner was working from home (he's on calls about 80% of the day), and we were doing no home schooling whatsoever. My child is only interested now in screens. Just whines from 8am onwards about wanting to watch TV.
Eventually I phoned the head and got a place as a key worker child. In the end she didn't need it because her year group went back, but it was close there.
Your kids are the very definition of key worker and vulnerable - they can get someone in to support him/her, we had to.
For September we've already asked grandparents to come down - we're going to share the rental of an airbnb with them, so they can be around from September - December. They are late 60s, so it's not ideal, but there isn't an alternative.
I honestly don't think there will be part time schooling in England in September, and even if there is, it will be VERY short term. So you've got short term options - stress leave and so on?