Nothing illegal about a sickness absence management policy that involves disciplinary action. If you had been there longer and they were to dismiss you because of such a small amount of absence, then you'd potentially have a case for unfair dismissal on the grounds that this response was disproportionately harsh. But you haven't been there long enough.
Even if they don't follow the procedure properly, your options are very limited - you would not be able to take them to a tribunal unless you felt there was some kind of discrimination, which it doesn't sound as though there is. Technically they do have to follow the statutory disciplinary procedure but in reality if they don't there's not an awful lot you can do about it at all.
Do take someone with you, a colleague or a union rep usually - they won't usually allow anyone else.
You mention about hoping DSs are not sick while you are there. I think the issue is if they are sick, for goodness sake don't phone in sick and use your sickness absence entitlement if this is going to be their response! It's certainly not going to make them any more sympathetic, as they might (should) be if all your absences were for you being ill. You shouldn't be off 'sick' if you are perfectly well.
If you hvae to take emergency time off to care for DSs for a day or two, you are entitled to do this, on an unpaid basis, see here. If you needed to do that for a day or two, they couldn't discipline you for it.