I just don't see why any sane person would take a small child into a toilet, assess it as too dirty and unsanitary for them to use (probably whilst the child is staring at all the willies hanging out) turn around, exit the toilet, assess the room to see if anyone looks obviously disabled and busting, decide noone else there will need to use the loo in the next five minutes, locate the disabled toilet (often in a different location) make sure it is unlocked and doesn't require a radar key, presumably there would then be another assessment to see if it meets your hygeine requirements before you deign to use it, then place your child upon it, presumably use it yourself, before you wash the child anyway and leave - hopefully without encountering someone waiting on the outside or walking past someone who has wet themselves and will now need to go home - and hopefully without having to deal with a wet child yourself as with all the faffing about who knows if they're going to hold it.
Or repeat above up to exit the toilet, find the ladies, announce your intention at the door to go on - risk someone objecting or complaining about you to the management - if they object you surely wouldn't enter anyway so what would you do - or just go in anyway, perhaps upsetting people in the process, presumably assess the sanitary facilities against the same criteria, before allowing your child to use them, then presumably using them yourself (unless you leave your child outside later to use the mens) then washing the child anyway and leaving, hoping there isn't a lurking manager waiting to ask you to use appropriate facilities outside, and gambling that you haven't actually upset anyone for whatever reason they are completely entitled to. From feeling threatened by your presence due to previous assault to being embarrassed or teenage or elderly or whatever.
How is this preference to just hovering the kid and washing them. Really. You can then inform the staff of the state of the loo and something might actually be done to help everyone.