"When you apply you must state you are applying for flexible working because you have a child under 5 and are applying to care for said child."
Nope. If your school are demanding this they are breaking the law. Anyone with 26 weeks' service can request flexible working, and they don't have to have caring responsibilities, let alone restricted to a child under 5.
"Your school (or council if it's a local authority school) will have 28 days to reply to your request"
Nope. Only if their policy says so. Employers have three months to give a decision. Normally it would involve a response/hearing earlier than that, but the law says 3 months.
"My head teacher told me that... schools actually don't have any valid business reasons for declining it"
Wrong, and that's ridiculous. There are plenty of valid reasons a school could use. Otherwise they'd have to agree stupid requests like can I work 5 hours spread over 5 days a week, one hour a day. No one would make that request but ridiculous to think schools are obliged to agree every flexible working request.
"Are there any other teachers part time in your school? If so then a precedent has already been set and they will struggle to decline your request."
Again, no. Each request must be considered on an individual basis, and although it might help your application if a very similar request in similar circumstances has been granted previously and you can demonstrate that it works successfully, there is no such thing as precedent having to be followed in this area.
OP the above is why you're far better off asking for employment advice in the Employment topic, rather than AIBU.