The answer in one documentary I saw is that Shakespeare was highly educated as part of an underground Crypto-Catholic community.
Shakespeare went to the local school, but didn’t go to university; OTOH, as you note, he was involved with the Old Faith in Britain, and was a teacher early in his adulthood, for people who didn’t want their kids going to gov’t schools, say, that kinda sounds familiar... anyway, it’s humbling to realize that, by the time Shakespeare was my age, he’d been dead four months. ;’)
Certainly Shakespeare was a country hick but a devout man with intelligence.
But Fulke Greville did have the high intellect and the breeding to bring any works of his time to life,thus making him the man to master many.
As far as underground Crypto-Catholic communities,I am sure many men of their day with traditional aristocratic breeding and specialist skills would be invited to join certain societies that are active to this day.