Let me add a few words to this thread, as this is a topic I know something about.
That Shakespeare was probably a Catholic has been virtually a truism among Shakespeare scholars since the time of A. C. Bradley (1851–1935). I was introduced to the thesis in the mid-sixties, when I was in college, through the then-famous edition (still very recommendable to students and casual readers, may I add) of the collected works prepared and edited by G. B. hαɾɾιson (1894–1991).*
Unfortunately, there is also very good reason to believe that Shakespeare had strong ɧoɱosɛҳųαƖ inclinations, upon which he regularly acted. This is discussed fully and frankly by the great Joseph Sobran in his brilliant contribution to the authorship controversy, Alias Shakespeare
If you can find a copy, my advice would be to get it and read it. It is both beautifully written and profoundly insightful. (I would offer to lend you my own copy, but as it is autographed to me by Joe, it won't be getting out of my sight till you know when!)
No one is perfect, of course. Despite hαɾɾιson's virtues as a scholar, he was sadly also a Vatican II enthusiast and a member of the committee
that crafted the first version of the Novus Ordo Missae
An admission against interest in the spirit of full disclosure: DAW hates this book.