Not quite sure where your getting that catholic primarys will take any kid, maybe in some city's that have larger schools or have a few will consider it but most areas only have one or two primany schools an don't have the room so catholic kids get the priority, it would be a rare occasion if a primany accepted a non catholic especially since like I say the school prepares them for their communion, most primanys are all close together so if you weren't catholic you'd just go to the Protestant primany school 3 mins away from it, half my family's catholic an they didn't have a mix in any of their schools, like I say high schools are different an will take a mix but it's only if they have room to do so, there's one catholic high school in my town (compared to the 4 protestant HS) an that covers the entire town, I know that most schools have now dropped religious studys but they are still named as being protestant, we don't have "state" schools here, it's all council an you only get catholic, protestant, or very very rarely non domain which isn't classed as either, then there's private which isn't anything but again those are rare, an any school will take disabilitys, none will ever turn kids away, most schools are capable an even have lifts for wheelchairs
I'd imagine England has quite a large diverse population, I think it's estimated there's around 55million living there, Scotland's a lot tinier with only about 6 million here, I know Birmingham has a massive diverse community