The actors talk without that awful universal period-drama accent, for a start, which really helps. It's like with Shakespeare- Tudor accents would have been strange and unintelligible to us anyway, so the voices are always going to be in translation to some degree. You may as well use patterns that ring true. They could just as easily all have been New Yorkers. I appreciate the colloquial language (down to the way the titles casually tell you that "it's 1535") in service of relatability.
Costume is another thing that's crucial- when actors look like they're inhabiting their clothes, when they look built to be worn around. Somehow the show makes Tudor clothes look fashionable, when usually they look only like costumes, doing the rounds between different shows and films (apparently in Jane Austin adaptations the same dresses pop up again and again).
Often too, the scenes appear to be lit by candles or natural light. This works to bind actor, costume and set together, kind of like a varnish or paint job, stopping you from seeing the actor on the set, the props on the table, as different things. This seems to be the hardest thing to make us believe- that the person actually belongs where they're standing.
It's tough though, because in order for it to be really authentic they have to shoot on real historical lawns and in real rooms, and (I'm thinking about the Birdman phenomena stuff still) these resist the fiction pretty stubbornly- the characters always seem somehow superimposed onto the locations in some way, like Snow White onto a Disney matte painting. It's not anything you can fix through light or costume. I can't help but admire the authenticity of the set, which means knowing that it exists on another layer to the action. Cathedrals are the worst for this, I keep expecting to see tourists.
In Shakespeare's theatre costume was the most important element, and there were no real sets. I wonder what a drama like this would be like playing on a completely bare set? If we saw an actually new Tudor building, would it look fake, too clean? I enjoy the arguments that spring up online about the conflict between what is truthful and what we accept as real on screen- the codpieces apparently are too small, but if they were bigger we'd think they looked weird. Also: are their teeth too good? Apparently not, since this was before the industrial revolution ruined us all. But even so, we see their teeth and think "historical teeth should be dirty, I'll suspend my tooth disbelief". I'm just thankful we don't have to smell them all.
Another funny intersection - Cromwell, in the show, posing for a Holbein portrait which still exists. Him holding still, chatting, being called away to other business. The show imagining outward from what we think we know, boldly and pretty cheekily encouraging us to believe that we can witness history. It's a pretty high stakes bet and it does pay off.