I’ve heard two different ways of pronouncing “Appalachian”: “a-puh-LAY-shuhn” and “a-puh-LATCH-uhn.” Which do you prefer? And where are you from?
Sorry, it ate my poll. Now I can’t add one.
This site suggests the latter pronunciation is used by the natives.
When I moved to central Kentucky, I was corrected, and the dictionary confirmed – it’s LATCH. In those days, dictionaries were still prescriptive, they didn’t care who they offended.
I grew up in NYC saying (not particularly often) a-puh-LAY-shun. After moving to Indiana, I heard the ap-PLATCH-an, or ap-uh-LATCH-an pronunciation for the first time. I looked it up, and found that the 3rd version is “preferred,” while the second is “acceptable.” The first time I met someone from the region, I asked her how she said it, and of course, she gave me the 3-syllable version.
What I mostly tend to say now, though, is ap-uh-LATCH-an.
I hear both, but prefer ay-pal-ay-cheein.
In New England, home of the Appalachian Mountain Club, it’s a-pah-lay-shun .
On some online maps, the top of my neighborhood is labeled “Appalachian Mtns.” So, I’m a “local”. 
App-uh-lay-chi-un. (The latter two syllables ~slurred together.) Don’t recall anyone using a “shi” pronunciation in my presence.
I grew up with the former and still tends to be my usual pronunciation, though I sometimes use the latter, as that is the more local pronunciation to the area.
Here’s an example of the “shi” pronunciation. The way that narrator says it (let the video play for a few more seconds to hear “Appalachian” a few more times) is how I grew up hearing and saying it. I try to say it more like “Apple-atcha” these days.
As I type this, I’m sitting in my cabin in the Blue Ridge Mountains and can see the Appalachian Trail running along a ridge top out my window. I pronounce it latch. The lay pronunciation is more common in the northern portion of the region.
The late great WV author Jim Comstock told the story that the Garden of Eden was located in West Virginia. When Adam and Eve noticed their nakedness, Eve said to Adam, “If you don’t stop looking at me that way, I’m going to throw this apple atcha.”
And that is historical fact of how Appalachia got its name. 
I live very near the Appalachian Trail and have often hiked sections of it. Around here (eastern PA just across the NJ border) it is the a-puh-LAY-shuhn Trail.
I figure the a-puh-LATCH-uhn version is more likely be heard farther south.
Definitely the “atch” pronunciation in WNC. It’s very clearly the correct appellation.
Growing up in Ohio, we pronounced it /hɪldʒæk/.
I pronounce it “APP-a-LAY-shun,” but I’m not from around there.
But, at least I don’t pronounce “Appalachia” as “APP-a-lah-CHEE-ah,” like Les Nessman. 
It’s ‘latch’ in my head. I can’t say I ever had a reason to say it aloud.
(Maybe I talked about it while reading the ‘Foxfire’ books)
Maybe I’m misreading the Latin-spelling pronunciations but I pronounce it æpəletʃən - app-a-laitch-in. The only word I can think of that rhymes with the third syllable is the name for the letter “H”.
That doesn’t seem quite the same as how I read the option in the OP of “æpəleʃən” which to me is pronounced the same as the word “Appellation”. I definitely pronounce the two words differently.
That said, when I use the word “Appalachia” without the “n”, I tend to use the southern pronunciation which seems to be æpəlætʃə and agrees exactly with what others have said.
Appalachia ends in New York State not that many miles from where I live. I’ve always heard the “ch” rather than the “sh” pronunciation.
Appalach. Kind of like “Apple-Latch.”