How did he grieve? Was his son buried in a catholic or protestant manner?
Unfortunately we don't have much, if anything, to record Shakespeare's immediate emotional responses to his son's death; what we do have is rich grounds for speculation in the form of Shakespeare's plays. The impact of Hamnet's death on William Shakespeare is often interpreted through the lens of the play Hamlet as it premiered a handful of years later, a theory generously helped along by their similar-sounding names -- but Shakespeare didn't invent the name of its principal character out of whole cloth (at least one Early Modern adaptation of Hamlet predated Shakespeare's, and has since been lost) and the precedent for Hamnet Shakespeare's own name is likely the name of an unremarkable Stratford-native: his godfather, Hamnet Sadler. Regardless, other biographers have remarked on apparent resonances from the name on down; it's possible for a Shakespeare biographer like Stephen Greenblatt to make an interesting case for a Hamlet-Hamnet connection, and for readers with a Romantic bent to scrutinize the plays through the lens of Shakespeare's real-life biographic narrative and vice versa. (Hamnet was also the fraternal twin of Shakespeare's surviving daughter Judith, and some see a resonance between Shakespeare's twins and the twins in Shakespeare's plays, for instance.) It would be hard for me to say with any firmness that any of this is a documentable fact, or that it's so likely that it should be taken into account by anyone who picks up Hamlet. Those possibilities are fertile ground for literary analysis and psychoanalytical readings of Shakespeare's works, both well and good in their own right but not compelling historical scholarship.
Hamnet was buried in Holy Trinity Church on August 11, 1596, as indicated in a parish register where he's listed as "filius William Shakspere"; the twins had been baptized at Holy Trinity eleven years earlier, and the church was (and is) affiliated with the Church of England. If Hamnet's death had been accompanied with Catholic ceremony, it would be effectively the smoking gun of William Shakespeare's own Catholic sympathies -- or at least those of the Shakespeare family in Stratford, given that William was likely absent at the time of his son's death, working in London -- but I can find no hard evidence of this. All signs point toward a fairly normal Protestant burial, whether or not Hamnet's parents and grandparents harbored Catholic sentiments in their private lives.
We don't know as much about Shakespeare's personal experience of grief as we know about, say, Ben Jonson's -- Jonson's "On My First Sonne" is quite frank about the emotions felt in the wake of the death of Jonson's young son Benjamin, as is "On My First Daughter", Jonson's poem about the death of his infant daughter Mary. Jonson and Shakespeare were hardly two of a kind in terms of personality, but if nothing else Jonson's works are a small window onto the landscape of parental grief among Shakespeare's contemporaries -- Elizabethan parents did grieve for their children, and this grief often assumed religious overtones as well as the more familiar registers of loss and affection. In the absence of any firsthand accounts or descriptions of Shakespeare's behavior at the time, we can't say whether he rushed to Hamnet's bedside in the last days of a fatal sickness or was blindsided by news of a sudden death while on tour, whether he felt guilty for not being more present in his children's lives or whether he experienced relief, whether he consciously or unconsciously drew on his experiences with parenthood and grief in the writing of his sonnets and plays. It would be neat if we did know, but the historical facts are so sparse here that all biographers who editorialize on the death of Hamnet are by necessity using their imagination to fill the gaps.
Some reading:
That's a shame - thank you for such an in-depth response and I'll definitely read what you've suggested. Always worried that i'd fall into confirmation bias if I interpreted Hamlet as being representative of Hamnet but maybe there is something more to it.
Thank you!