I guess until recently I rarely heard the word and associated it with “arse” as opposed to just “ass” with a British twang applied to it.
Is it pretty much pronounced “oss”? That’s what I’m hearing here:
Depends where you are in Britain. In the south it’s ahhss.
Most British dialects are non-rhotic, so no, the R wouldn’t normally be pronounced as a consonant. If you’re familiar with the International Phonetic Alphabet, the usual pronunciation would be something like /ɑːs/ – your “oss” is a reasonable transcription.
Of course, if you’ve got an American accent, then the R is definitely sounded: /ɑɹs/
The R is pronounced in Canada by many speakers, particularly those with strong recent ties to the UK, such as Newfoundland. Such as the expression “arse-foremost”, which means “backwards”
In Britain, “shite” is also pronounced to rhyme with “white”.
How else would you pronounce shite? 
The R would be pronounced in both Lancashire and the West Country
The Cambridge Dictionary in fact gives that as the UK pronunciation.
“Ass,” however, is pronounced differently ( æs, not aːs) even when it means butt.
For certain values of West Country, yes - that is, the oi be drinking cidarr accent everyone thinks is so prevalent.
Mostly this. It’s not that the R is specifically silent in “arse”. In fact, the R does have an effect, because “ar” makes an “ah” sound (as “ah” is pronounced here). So, in my accent at least (mostly RP) it rhymes with “pass” (pronounced “pahss”) (and “saw” and “sore” sound the same).
That may be confusing, so alternatively, it’s a bit like “Hahvahd”.
In Ireland, the “r” is pronounced.
I’m not sure that’s true.
I’m not even sure it’s true if it was amended to ‘most English dialects’. (Non-rhotic accents are mainly found in the South and South-East of England).
In terms of population, I would guess the clear majority of native English speakers in England are non-rhotic. The 20th Century Wikipedia map, where London, Birmingham, and Manchester are all in non-rhotic areas, would seem to bear this out.
If “arse” is largely pronounced without the r sound and in the US the word is ass, which is definitely without an r sound, whence cometh the difference. Is the original word “arse”, and those who came to America came from the r-less pronouncing areas and later changed the spelling?
AFAIK the common American spelling is a bowdlerisation. They didn’t change the spelling to conform to pronunciation; they changed it to try and make the word a bit less offensive, so that in calling someone an “ass” you could be seen as comparing him to a donkey rather than to an arse.
“Ass” meaning donkey (and with a short a) is an insult in Britain too, but not nearly as strong a one as “arse”. A (British) ass is a foolish, but mostly harmless and ineffectual person. An arse is a very unpleasant person, one who enjoys making other people unhappy (or, at the very least, does not give a damn if he does so). As I understand it, the American “ass” is somewhere in between.
In the midlands of Britain ‘arse’ rhymes with ‘farce’. Although I guess Americans pronounce ‘farce’ differently…
Yes, that’s what I was thinking, though in my rush to reply it’s not what I ended up writing. 
I actually don’t know what proportion of British dialects are rhotic, not least because the whole concept of a dialect, even more so than a language, is nebulous; if you try to count them you can get wildly different numbers depending on how granular you draw the lines. However, for any given speaker it’s (usually) not hard to characterize their speech as rhotic or non-rhotic.
It’s not that the r sound isn’t pronounced. As mentioned upthread, an English person with a non-rhotic accent will pronounce “ass” and “arse” differently.