It’s not unheard of even today. There was a famous British cookery writer and television chef named Fanny Cradock.
In Ireland it seems to have gone out of fashion. I can think of two times in my adult life I’ve heard it although it was very common when I was a little boy in the '80s.
On some sort of unified scale of rudeness, it’s somewhere around the same level as ‘dick’. Not quite as rude as ‘pussy’, IMO.
Two nations, separated by a single language. ‘Fanny pack’ sounds like a sanitary device, to UK ears. Funny really.
I had a teacher who considered “Sweet Fanny Adams” a little too racy, and used the ‘euphemism’ “Sweet Felicity Arkwright”.
As kids we used to kill ourselves with laughter at the mention of “Aunt Fanny” in the Famous Five novels by Enid Blyton.
I suspect that a proper fanny didn’t evolve until at least when out ancestors started walking upright - most likely in high-heel sandals I would imagine.
Let us not forget that she did her cooking show with her husband… Johnny.
(For non-UKians, that’s a euphemism for condom.)
The author Fanny Flagg gets the odd titter as well.
There is a story about the filming of Cleopatra, starring Elizabeth Taylor and Richard Burton, stating that the American director told the British actor Burton to “Grab her by the fanny!”, which he did.
The 1835-40 cite was in Speaight *Bawdy Sngs of Music Hall, 76.
The US fanny in 1919 was from something called 12th U.S. infantry
…Or that BBC presenter Frank Bough once (allegedly) said in a link to her show, “I hope your doughnuts turn out like Fanny’s” 
Bum Bag - never used bum pack…
Famous English cook in the 70’s was Fanny Craddock
You don’t say?
Remember the lyrics to the theme song of The Nanny?
She was working in a bridal shop in Flushing, Queens,
'Til her boyfriend kicked her out in one of those crushing scenes.
What was she to do? Where was she to go? She was out on her fanny…
Trust the Brits to drive on the wrong side of the road.
Any Brits I host in Helsinki tend to get a giggle out of a local restaurant next to Sinebrychoff Park. The food is good, it’s great for brunch, and the atmosphere’s nice but…well.
That reminds of when one of the king of Sweden’s former girl friends started a restaurant in, I think, London and named it after a pet form of her name Christina: Titti’s.
Not in my experience. As a female child, if I’d said “fanny” in front of my parents, I’d have expected to be told off for saying a rude word. Both it and willie[1] were our childish swear words.
I don’t think we knew the grown up versions, so to us these were the height of rudeness, and yes, if a teacher had to read them out, the class would dissolve into hysterical laughter.
[1] Penis. We loved the film Free Willy over here, oh yes 
Ma’s out, pa’s out, let’s talk rude: pee poo belly bum drawers.
Out of interest, what were you told to call yours? I think my sisters were told to use fanny (1970s) as opposed to other friends I’ve discussed this with who used “foo foo”, “patata” (Italian for “potato”), and “penol blein” (Welsh for “front bottom”)!