Yeah, and thus no C of E; would there have been a Protestant movement, much less one sponsored by the Crown? If England had kept on having Catholic monarchs, would it have kept the monarchy system at all or would the royals have toppled? Pretty much everything about the UK would be very different indeed! Interesting to think about.
Another King Arthur would be neat. 
Oh we studied lots of history - years and years on the industrial revolution, the First World War, the rise of Nazi Germany, the Second World War, the Cold War. But studying the reigns of English kings from nearly a thousand years ago is considered, rightly in my opinion, not to be worth going into much depth over - there’s so much of it, it’s all so complicated when you go into details (the Magna Carta for instance was signed, then renounced, then fought over, then reissued and signed again, then amended repeatedly and resigned, etc) and it’s all quite literally ancient history. There’s a period in English history of about 500 years that I never studied formally at any age, and I can’t think of any time I’ve felt disadvantaged for it.
So yes, I think for most people in this country Richard I would simply be ‘Richard the Lionheart, a brave guy who lead a Crusade’, and King John would be ‘a bad king who signed the Magna Carta’ or ‘the bad guy in Robin Hood’, if those names even rung a bell at all. I’m sure a lot of people know lots more about them, but I think that’s an accurate assessment of what the average person on the street would know.
I didn’t learn about any of that in school. IIRC I had two years between middle school and “options” (the point in British education when teenagers choose a number of subjects to study to examination level), during which History was compulsory. I didn’t opt for History (not a popular option, as I recall), and I only have hazy memories of what we studied in those two years. I can remember learning about the Norman invasion, and ancient civilisations such as those of Mesopotamia, but not kings and queens of England/Britain, or later British history.
Well the 3rd in line to the throne is a Henry so if Prince William dies childless (or marries a Catholic gal) you’ll get your wish. 
I blow my nose at you, so-called Arthur-king, you and all your silly English kaniggets. Thppppt!
Come to think of it, there hasn’t been a King Richard for a long time. Does Richard III trump Richard the Lionheart in the mojo department?
It does, especially since in hindsight Richard the Lionhearted does not really seem that appealing when you consider that he spent more time in France than in England, was really an Englishman in name only, and spent a ridiculous amount of time and money on - as Sir Hiss, voiced by Terry-Thomas in Disney’s Robin Hood would say - that crrrrraaaaaazy crusade which was ultimately a failure. Richard II was also not a very good or well-liked monarch.
Also, nobody wants to deal with the blitz of dick-related puns and nicknames which would inevitably result in the media.
Well, maybe. But we had Edward VIII and not King David, and we had George VI and not King Bertie, so even if Harry did succeed he might not reign as Henry IX.
Edward VII’s first name was also Albert and he went by Bertie as well. And what’s really funny is that George VI’s younger brother, the Duke of Kent’s first name was George!
And even Queen Victoria’s full name was Alexandrina Victoria and as a child she was called “Drina”.
Probably. Martin Luther and the Protestant Reformation had started some years earlier than the separation of the Church of England, and was a force to be reckoned with throughout Europe. Likely, it would still have taken root in England in some form. The Puritan movement started out reforming the COE, but France had the Huguenots with a Catholic king on the throne.