My understanding is that Shakespere had Catholic sentiments but one had to be very careful during those times. Some Catholics would make a show of going to Prot church but would remain secretly Roman.
Years ago, after studying his Second Tetralogy (Richard II, Henry IV, Pt. 1, Henry IV, Pt. 2, and Henry V), I got an impression similar to Roscoe's. Shakespeare seems to have had an intense inner conflict regarding the authority of the Tudors who had politically "overthrown" [for all intents and purposes] the greatest authority known to English Christendom, i. e., the Roman Pontiff, the same authority that had tacitly sanctioned the reign of the Tutors and previous English kings [for, up until the English "Reformation," they were crowned according to the rites of the Roman Pontifical by an Archbishop whose jurisdiction was given by the Pope]. If such a great authority can be so easily demolished, what stability, then, could the authority of the monarchy itself have or claim to have.
This impression I felt the strongest after I read Richard II, which is ultimately a lyrical and theatrical commentary on the question of how lawful authority is determined and the complex ramifications of a bad ruler and subsequent usurpation.
He availed himself of tropes, metaphors, allusions and archetypes that hearkened back to English Catholic heritage, but this may have been merely a vestigial cultural marks that the Anglican transmogrification of England could not erase (nor did it ever, just ask the "High-Church" Anglicans). Though I would not readily dismiss these as evidence that he had some Catholic leanings.
I think he was a very conflicted individual, especially when it came to politics and religion. He may have had a nostalgia, a vitiated certainty, or even a conditional eagerness to embrace the faith, but he never showed in his writings the fruits usually associated with a supernaturally infused faith.