I (or my partner) have always taken annual leave if DD is sick and cannot go to nursery.
AIBU to think that if your child is sick and cannot go to preschool/nursery you should take annual leave to look after them, rather than claim you are working from home? Especially if there is no issue with you having run out of annual leave etc.
I just couldn't do my work properly whilst looking after/dealing with a sickly child.