Dealing with parents who don't seem willing to have adult conversations in order to try and navigate these situations.
Navigate a period starting?
Do teachers need to be told that young girls will have periods which usually happen once a month for a round 7 days and may need to use the bathroom more often. I was not aware teachers did not know this 
Teachers are explaining very good reasons why they simply cannot allow every child to go as soon as they ask
Nope not good reasons at all. All i have read is "well bad children abuse it and it causes disruption that I will then have to deal with so no children can use the toilet^
how dare you? My child must be allowed full and unfettered access at all times".
Yes how dare you.
How dare you strip my child of their dignity and humiliate them in front of their peers. How dare you put them in a position where they are open to being bullied.
If every parent whose child had a specific need arranged it with school then that would cut down on the requests.
Why not just put the girls in menstrual huts 
Its a period not an illness.
What strategies do your children use during times when toilet access is severely limited eg during public exams (surely they aren't allowed to go twice during a 1 hour exam as a PP said her daughter needed to go) or maybe on a train journey (no toilets on tube trains)? Are there ways that you help them to manage these situations and if so would they work at school?
Most trains have toilets.
We dont have the tube in my city not everywhere is London. Most bus journeys are 20 minutes and i would imagine they would seek out a toilet immediately after getting off the bus.
They are 11 and not doing exams.
School is not a toilet free zone like a bus or train. It has the facilities to enable young girls to clean themselves yet teachers like you would rather deny them that or else you may have to actually manage your students effectively.
Yes the strategy is if you need the bathroom then find one asap.