There are still some parish churches that do a lot to promote children's music and development, and encourage aspiring organists. An amazing friend of ours started a children's choir at our local church which gave our children their first experiences although they both moved on to choirschools. Most of the others involved have remained state educated but several joined more serious choirs and I suspect there may be other potential choral scholars coming through. One older boy learned organ and is now studying music and conducting seriously.
I agree that public schools and ex-choristers probably dominate the higher level / more serious choirs, perhaps more so for young men. Only a small number of colleges have choirs that are only recruited via the choral awards though, many have additional auditions during first term and the choral scholars' role is leadership within the choir. And plenty take all comers (often including university staff). So a lot of young people have the opportunity to experience chapel music while at university and find out if it is something they want to do more of.
This year I wouldn't be surprised if the choral applications were/will be (Cambridge doing their auditions after offers) more dominated by independent school applicants than usual, as the open days normally held late April/early May were of course cancelled. Some who might have been persuaded to have a look and encouraged to have a go won't have had that opportunity this year.
Organ scholars remain a niche interest. Those that for a college with fewer services are often responsible for recruiting and training the college choir, choosing all the music themselves, and sometimes in charge of running college music in general! So those posts attract a fantastic musician who is also a great organiser.
My son's at the more serious end; we knew he loved pipe organs at the age of 4, encouraged him to try the piano out and got him lessons at 6. He was ready to start organ lessons at 10 although practice time was limited while at prep school, now he's a boarder and can practice more his playing has really taken off. He won't be doing grade exams on organ! He is singing in his school's chapel choir and is likely to have opportunities to accompany some services there.
A current organ scholar I know fairly well started as an excellent pianist but added organ as a teenager. He's in one of the more serious / career oriented organ posts.