Personally, I'd have filled in the formal request ages ago (they have so many days to respond to your request), you can then appeal it if you want to. I see you've spoken to ACAS now and I'd do as they suggest.
I expect you can't go straight to discrimination if you've not had a formal flexible working request turned down. However they will need to justify why they can't give you the days off you're requesting, what is the impact on the business of you being off that day? If you can work out what might be the problem with that day, you can put some suggestions in your application about how you can make it work. Plus once you have their reasoning in writing, you can work at addressing it or taking it further if it's discriminating.
Also, how long are you asking to work part time for? I found my request for flexible working (3 days, Mon, Wed & Fri) was partly agreed because I wasn't off Mondays and Fridays, and also, I asked for it for 6mo, moving to 4 days a week after that, and finally back up to f/t. My boss was happy with this as she didn't want a permanent reduction in hours, I never would have got a permanent change.
Annoying as it is, it isn't their problem your nursery of choice is full on the day they want you to work, they don't have to allow you to come back on p/t hours (if they have a business reason why not).