Cunt is almost certainly Germanic in origin. Old Norse most likely.
For the UK specifically, it likely comes from the Latin variant cunnus, meaning vulva. And Romans didn't spend a great deal of time north of modern-day Newcastle, a fair bit south of the border.
Were they not allowed 1 swear word in each film to keep the same age rating in the US? Of course probably different for other languages, but I think this would be a good case to make use of it.
The actors who played Merry and Pippin discussed this in a video once, and went back and forth on where they'd put it. I particularly liked a suggestion in Moria: "I'm so fuckin' sorry."
But the best reply to this idea is Boromir at the council: "The trilogy has no f-bomb. The trilogy needs no f-bomb."
If we're talking Two Towers I want to save the f-bomb for when Aragorn tosses Gimli at Helms Deep. Gimli just roars "Fuuuuuuuuuuuuck!" as he flies through the air.