So it's fairly obvious that when speaking English in 17th century London, the "r" was most definitely pronounced. But what of the late 18th century, the time period portrayed by the Brigade of Guards?
Well, that's where things start getting interesting. Even in 1757, RP wasn't around when Dr. Johnson wrote his A Dictionary of the English Language (although he did acknowledge there was a wide range of existing accents at the time). In 1780, English writer Thomas Sheridan remarked that the /r/ "always has the same sound, but is never silent." It wasn't until the 1790s that things really seemed to have changed. In fact, Americans returning to England after the War for Independence were reportedly surprised by the changes in fashionable pronunciation. By the early 1800s, John Walker, in his Principles of English Pronunciation noted how the times were changing when he commented that:
[R] is never silent…In England [R] is never silent … In England, and particularly in London, the r in lard, bard, card, regard, etc. is pronounced so much in the throat as to be little more than the middle or Italian a, lengthened into laad, baad, caad, regaad … if this letter is too forcibly pronounced in Ireland, it is often too feebly sounded in England, and particularly in London, where it is sometimes entirely sunk …
Sheridan's remarks seemed to imply that non-rhoticity was undesirable at the time, but nonetheless it soon became the norm. This might make sense because linguists believe it may have originated from the East Midlands before traveling down to London and gaining traction there. At some point, this burgeoning trend to drop the "r" became the fashion rather than the faux-pas. Therefore, linguistically, one could say that RP developed from a changing London accent that incorporated elements from the East Midlands, Middlesex, and Essex.
RP then really came into its own in the 19th century as a socially exclusive accent being taught in elite boarding schools and at Oxford and Cambridge. The actual term for RP was coined in 1869, and it received wide recognition in 1924 when Daniel Jones adopted it in his second edition of the English Pronouncing Dictionary. Two years previously, though, the BBC adopted it as their broadcasting standard.
Indeed, one school of thought holds that the accents of America, Australia, and Canada are "snapshots" in time for an evolving English dialect. Sure, each went on to develop their own regional accents and dialects, but their overall nature is a reflection of the point in time they left the "mothership" of jolly old England. For Americans, we retained the generally rhotic accent of pre-Revolution, while Australians (who were colonized after the accent shift to non-rhotic) generally do not. And Canadians, well, they're somewhere in the middle of it all.
As for us in the 4th Coy - we like the put on a show as much as the next King's man, but let's not sweat our R's too much...
Fisher, John Hurt (2001). "British and American, Continuity and Divergence". In Algeo, John. The Cambridge History of the English Language, Volume VI: English in North America. Cambridge: Cambridge University Press. pp. 59–85.
Crystal, David (2005), The Stories of English, Penguin