While watching an episode of Sharpe from the 1990s, I found myself wondering how different the upper class Englishman accent would be in the actual Napoleonic timeframe. Would the British upperclassmen have the typical sort of Downton Abbey wealthy accent? If not, what did they sound like? What about French, German, or Scandinavian languages? What about spoken English found across North America? We can see clearly in period writing that it differed a fair bit, as would the vernacular of the day. But how different?
You can find various different videos of people doing Shakespeare in "Original Pronunciation", so you can hear what English would have sounded like a few hundred years ago, before Napoleon, but it gives you an idea of how different accents can change.
The side by side comparison begins at about 3 minutes.
Slightly after your time period, but Sam Weller from Dickens' Pickwick papers (1837) is written with a very strong cockney accent, one of the features of which is exchanging the 'v's and 'w's ('wery' and 'Sam Veller'). This is something that's disappeared from modern London accents.
The Concise Oxford Companion to the English Language (no idea why it's online) has a 1791 quote from an 'elocutionist' called John Walker, who identified four faults with the cockney accent: pronouncing words like 'fists' or 'posts' as 'fistiz', 'postes' etc., swapping 'v' for 'w', pronouncing 'wh' as 'w' so that 'where' and 'were', 'while' and 'wile' etc. become homophones, and dropping the 'h' or putting it where it shouldn't be ('art' for 'heart' or 'harm' for 'arm').
If you compare that to modern accents, the extra syllable in 'fists' apparently disappeared in the 1950s, the 'v' for 'w' disappeared shortly after Dickens was writing, the third 'fault' has pretty much become universal (among English accents anyway, a bunch of Irish and American accents still maintain a distinction) and the fourth one is the only one that's still treated as a mistake.
I don't really know about what you're really asking about (the history of RP) but there have been a couple of posts about it on this sub before (I'll link when I find them), and from what I remember it developed after the Napoleonic wars. It's interesting though that, even though he points out those 'faults', Walker still says that Londoners are 'upon the whole the best pronouncers of the English language' and that they form a model of pronunciation to provincial areas. This seems to indicate that the prestige dialect was the London one, even including a number of features that later became lower-class.