What is the name of the kind of accent the classic "butler' or servant-of-old-money families called?

“Received” English. Also known as BBC English.

The accent is immaterial. It is command of the language (whatever it is) which matters. That and the understanding that true class is something money can’t buy. Pretention IS tolerable to a point, but a GOOD butler (as with his employers) knows quality when he sees (or hears) it, and this is particularly true in the matter of “quality” people. A prize butler will recognize a gentleman though he appears at the doorstep dressed in rags speaking a language the butler has never heard! It’s the sort of thing impossible to accurately explain, but fundamentally it is the reason that Nelson Mandela was a gentleman while Trump is unequivocally NOT!

I’m giving you a Good Answer point because your answer went above and beyond excellent.

You were SO GOOD until your last line. Trump sure isn’t old money classy. Agreed. But Nelson Mandela is a terrorist who burned people alive. Leave it to Obama to take him off the list. Terror can never be class, no matter how it disguises itself.

That caveat aside, excellent answer

The great tragedy with you & the wolf is that you are so clueless that I have difficulty believing you authentic. I actually prefer thinking you Russian plants here, but the fallacy in that line of reasoning is that you are just SO bad at it. But if you aren’t Russian tools, you are both so lost that it is empirically unfair for me to even discuss the most basic of issues with either of you. Obama took Mandela off the list? THE WHOLE WORLD TOOK HIM OFF THE LIST and placed him at the TOP of the list where he damned sure belonged. The fact that you cannot see that leaves me dumbfounded. By the same token, the fact that Trump is the exact opposite of a gentleman requires no elevation in class to be recognized in a heartbeat. YOUR DOG would know it and probably leave the room upon his appearance.

A posh accent.

I disagree with you too @stanleybmanly. Any man that uses words like “bigly” and “hamburder” must be a yugely stable mental giant.

[Mod says] Moved to Social on request.

The vast majority of old school butlers served as lower ranked officers in the military where speaking posh was the norm.
We on the other hand hire any old Tom Dick or Harry, so long as they’re loyal & cheap.