I usually say, “the cat on the reeds.” I’m a hep daddy.
I once actually heard a TV commentator–broadcasting LIVE–pronounce the word as “CACK-a-phony.” I was astonished, and about 10 seconds later, my sister, who was watching the same program, phoned me in disbelief. “CACK-a-phony?” was all she could say before we dissolved into fits of laughter. 
I had never heard that term before in my life and had to look it up. Now I feel like an idiot because I probably SHOULD have known it. It means podiatrist. My boyfriend is halfway through podiatry school. Surely this word should be in my vocabulary, no? But seriously, never heard of it until I read this thread. Initially I assumed it would be related to chiropractors.
po-DIE-a-trist
But if I had to say “chiropodist,” I would probably say KY-roe-pode-ist.
Hugh Grant pronounces it sax-OFF-onists in About a Boy (I think the clip is in the deleted scenes).
No, vi-o-LIN-ist. But the stress is already at the end of vi-o-LIN and I don’t think, OTTOMH, that we move stresses backwards in formations like this. On the other hand with a longer syllable at the end, we go from ARQU-e-bus to arqu-e-bus-IER.
It’s a shibboleth, intended to keep Americans from passing themselves off as English.
My point about violinist is that the first part (violin) is unchanged. You pronounce violin and violin(ist) the same.
Here’s a thought…how do people here pronounce ‘pianist’? 
I’ve heard both “PEE-uh-nist” and “pee-AN-ist”. I think both are acceptable. Checks . . . . Yep.
And I always said “saxophone-ist”. I’ve never heard of a “chiropodist.”
ETA: American, BTW
Nobody’s brought up flutist/flautist yet?
“Saxist.”
Har har. I don’t think I’ve ever had occasion to say the word, but if I did I would say SAX-uh-phone-ist.