This type of leave used to be described as for emergencies, but that word isn't used in the legislation and case law has clarified that it doesn't need to be an emergency, just unforseen. The actual legislation says:
Section 57A, ERA 1996
"(1) An employee is entitled to be permitted by his employer to take a reasonable amount of time off during the employee's working hours in order to take action which is necessary -
(a) to provide assistance on an occasion when a dependant falls ill, gives birth or is injured or assaulted,
(b) to make arrangements for the provision of care for a dependant who is ill or injured,
(c) in consequence of the death of a dependant,
(d) because of the unexpected disruption or termination of arrangements for the care of a dependant, or
(e) to deal with an incident which involves a child of the employee and which occurs unexpectedly in a period during which an educational establishment which the child attend is responsible for him.
(2) Subsection (1) does not apply unless the employee -
(a) tells his employer the reason for his absence as soon as reasonably practicable, and
(b) except where paragraph (a) cannot be complied with until after the employee has returned to work, tells his employer for how long he expects to be absent.
As you can see, the right to time off is not simply to care for a dependant, unless that person falls ill. It is designed to cover unexpected disruption to care of a dependant. In your case, that disruption to your childcare is no longer really unexpected, so I am afraid that this right is not really the correct one to be using for a more long term arrangement like the one you are suggesting. If you can fix a day a week when you need to be off, it doesn't really fall under this legislation. If your DH continued to use this right and his employer felt it was inappropriate, it could unfortunately potentially become a disciplinary situation. If he took time off which had specifically been refused, it could even be a termination offence.
If your DH wants to request a permanent change to his working arrangments, he could use flexible working rules. Otherwise, I am really sorry but I'm not aware of any real legal grounds to make changes. Realistically, maybe you could agree one day a fortnight off. I know that's not as good as you hoped, but if that's what you can financially manage, it might be the only option.
I have to say, if I was your employer, I would also refuse the working from home request. Most employers will take the view that you can't be responsible for care of dependants and, at the same time, do a full day's work. It's the same reasoning that means that women who ask to work from home after returning from maternity leave are generally refused if they don't have childcare.
Sorry, that's not very helpful. Can I think of anything else.... Hmm, could your DH use some holiday to add to the fortnightly days off?