I can't get the link to work either, so really don't know the details.
I am unintentionally home ed ing, children with disabilities who fell apart when mainstream wouldn't accomedate them. Now trying to slowly integrate into a much much better (although worse according to ofsted) school with a fantastic approach.
Surely the rule of thumb would be the resident parent, or the parent doing the caring day to day gets the say- the same way you would with a preschooler- so your choice whether you attend toddler groups or go to the park or use tv etc is up to the person watching the kid then, same way resident parent has no say when non resident parent takes the kid out with new gf etc.
But I don't see why you couldn't look for a middle ground when dealing with a little kid. Curiculum isn't really much other than playing & a bit of structure & phonics etc at reception age. I have done and am now again doing part time (yr 2 cyrrently) and it works just fine. I am very boring pressure with academic stuff and school are find with this (although part of that is all our professionals involved support this). We got swimming/horse riding/forest school/park/libary around school hours and pay a tutor for one hour a week also. Other than reading lots and providing lots prewriting activities and talking about numbers in general we just concentrate on having fun and managing their disabilities.
Good luck sorting it out