I’ve heard several different ways but I am not sure I’ve ever heard anyone authoritative say it out loud. I think what I’ve heard most commonly is “Roy-ters” (which actually doesn’t even make sense with the spelling).
Royters
The same way a Dubliner pronounces “writers”.
James Joyce and GBS doesn’t sound right to me/
Roiters does tho’
Why doesn’t it make sense? The name is of German origin, and in German “eu” is pronounced “oy”.
The original Reuter was German and that (minus a bit of English accent) is the regular German pronunciation.
ETA: Ok, I’m a bit slow today.
Royders
I still incorrectly say it “Rooters” a lot of the time.
It is indeed Royters. The have been a client of mine in a professional capacity for over a year and we discuss them and consult for them in mega-corp meetings every day.
OK, this is GQ, and what I have is half dim memory, and half WAG, but here goes:
I remember being taught that in German, vowel combos take the sound of the second vowel. That would make Reuters sound like ‘Rutters’. Now, to an Englishman or American, that would sound like someone rotten, or someone copulating in a vulgar, animal-like manner. They may have shifted the vowel sound (had a vowel movement?) to avoid the negative connotations when they went international.
Ha, no you are NOT slow. I just opened this thread and was busy thinking that of course the pronunciation makes perfect sense, but I had the vague notion that the original Mr. Reuter was Swiss, for some reason, but now I have checked and he was apparently born in “Cassel”. Is that right, and if I now wonder why it is not spelled “Kassel”, am I being slow?
If permitted, in G.Q, though, I want to say that I really like jjimm’s answer. 
I’m not a native German speaker, but I’m pretty sure this isn’t true. As far as I know, “eu” always makes the “oy” sound (e.g., Freud). You might be thinking of “ie”/“ei” - “lieben” pronounced “leeben,” “leid” pronounced “ly-d.”
Nah, the “sound of the second vowel” thing is a useful mnemonic more for the “ie” and “ei” combinations rather than for this one - helps one distinguish between Wien, the city and Wein, the nice happy drink - that sort of thing. Of course it also meant that my friend Sheila, when living in Germany, met a LOT of people who were determined mnomics tend to have use but limited use: just think of how many native Anglophone people have only been further confused by being advised about the " ‘i’ before’e’ except after ‘c’" thing.
This isn’t true. There are exceptions. For instance - *Haus * is pronounced (nearly) the same as its English equivalent, “house”. The “eu” sound is another example.
But “eu” *always * sounds like “oy”. It isn’t a special rule for just this one word. *Neu * (new) is “noy”. *Europa * (Europe) is “Oy-ropa”. And so on.
Northsiders or Southsiders?
This is true for the ei and ie combinations, but eu is pronounced “oy”. cf “neun” (“noin”, nine) and “Deutsch” (“Doich”, German).
Ah, I missed the edit window. Oops.
Here is what I would have written had I not missed it, not that there is any of great importance, but merely that I don’t like to see that I have written utter sentence-missing gibberish.
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Nah, the “sound of the second vowel” thing is a useful mnemonic more for the “ie” and “ei” combinations rather than for this one - helps one distinguish between Wien, the city and Wein, the nice happy drink - that sort of thing. Of course it also meant that my friend Sheila, when living in Germany, met a LOT of people who were determined that her name could not possibly be pronounced “Sheela”.
Mnemonics tend to have use, but limited use: just think of how many native Anglophone people have only been further confused by being advised about the " ‘i’ before ‘e’ except after ‘c’" thing.
[Tried to edit to make a bit more sense. Sorry, I might have hit “submit” instead of “preview” or it might be the work of whatever gremlins were keeping the board been down earlier today, but I do not know which. However, my first version just had sort of half a sentence missing, but no real change was made to the meaning of the posted message, if that’s all right .]
It’s the same city (btw. I lived there 1995-2000.) In the oldest extant written records from 913 it’s called Chasalla/Chasella, probably derived from latin castellum (fortress). In later centuries this became Cassel. In 1926 the city was renamed Kassel. The reason for this last change is that “C” on its own - not as part of “ch” or “sch” - is unusual in German. Most of the it can be traced to a latin or other foreign origin. Today it’s almost exclusively found in relatively recent foreign words or proper nouns. The spelling of many words that included such a “C” in the past was germanized at some point, substituting either “K” or “Z” depending on the pronunciation.
Thank you, Celyn and Aktep for slaying my beautiful theory with ugly facts. Er, I mean fighting my ignorance.
Reuters, the lot of you.
Wirchlich, nicht gut bei mir, solch ein Fehler hier zu schreiben
I played a reporter from that particular news service in a high school production of Inherit the Wind, and was told - correctly, I believe - that it’s pronounced “ROY ters.”