That sucks, OP. Is there any chance at all of alternative childcare? Also i think it would be a good idea to call HR (assuming you have one since it's a big company) and tell them what happened. See how they react.
The employment of women is a huge problem area but as long as the UK keeps voting in governments that simply don't care and nobody (ie the government) is forcing companies and society to cut the sexist crap this will continue.
I have a list as long as my arm of interviews i've been to where the illegal questions have been asked (I'm not in the UK). Some i have simply walked out of with a "this isn't going to be a good fit", some i have asked them wtf they're thinking of asking me that and had they asked all the men? Twice i got my union involved and once it went to court. It is never fun.
The average, i think, is that most children have 2 parents who are together, single parent families still (probably just about) in the minority? So what we should be doing is making it socially unacceptable for families where there are 2 parents together that the fathers aren't stepping up. And yes their jobs are often so fucking important, if only because they earn the most but why are we there?
Where i am each parent is entitled to 10 sick days (with full pay - covered by the health insurance, which is nearly but not quite like the NHS) per year per child (so in total each child gets 20 days entitlement) which is additional to your own sick days. It seems immensely unfair if a mother calls in sick because they have to take Tarquin to the doctor if the boss then says "can't your partner do it?" but until we can be pretty sure that that discussion has been had at home, how else can we make people think about it?
When my DC were at school i had a childminder who cost a freaking fortune, nearly the marginal amount we got from me going back to work after ML, and a DH who worked split shifts. They were the contacts the school had, and i was 3rd. The first time the school secretary called me, i went to collect the DC. And asked, pretty casually, if they hadn't managed to reach the other 2. And the reply meant a very stern discussion about following the list because I'd had to drive 40 minutes (the fact that I'd had to leave a sales meeting wasn't mentioned but it was a huge irritation to me). Why the discussion "well, IME" says secretary "sick DC want their mum". Well i want a bloody Aston Martin DB5 but i can't have one.
And so on and so forth. And then when it has finally sunk in that both parents should be playing a role - life may get easier for single parents (who we need to give a bit of extra support to, it's the right thing to do).
Hold MPs to account. Hold HR to account. Join a union. Get DH involved where possible.