I’ve almost always heard it and refer to it as “woostisher sauce”. Grew up in Arkansas, parents from Oklahoma and Texas.
Here, let me help:
This has been coming up enough here at SDMB that I bookmarked that just for me.
So given this and the other comments, either we have a Yorkshire/non-Yorkshire difference, a North/South Difference, or an A. Gwilliam/everyone else difference. Hm! 
I’m leaning to the Yorkshire/non-Yorkshire thing at the moment. Why would one county name have a different suffix, unless you were using that name much more than the others?
If you say “Yorkshuh”, how do you say “Hampshire”? (I ask about that one since the “root” also only has one syllable.)
It’s already been pointed out on this thread that there very many accents and dialects in this country. They change in the space of just a few miles. I come from Coventry where we have a different accent to the residents of Birmingham (twenty miles to the west) or Leicester (twenty miles to the north-east)
So I can well believe that the residents of one part of Yorkshire pronounce the name of their county differently from residents of another part of the county.
/hæmpʃə/. Rhotic British accents would add an r to the shire part, but it’d still be a /ə/. Occasionally you might hear /hæmpʃi:/ but only occasionally if the word were in complete isolation, like answering ‘where are you from?’ Same with all the shires.
Oops, I meant /hæmpʃɪə/.
The same. I say all the counties ending in shire as -shuh
Yeah, sounds like Americans vocalise the “shire”.
I’ve lived in Hampshire almost all my life, and the answer is “It depends.” In casual speech, ['hæmpʃə] (but with an intrusive “r” before a vowel, so ['hæmpʃər ənd 'sʌri] for “Hampshire and Surrey”.
If I’m emphasising the name, I’ll pronounce it more like ['hæmpʃɪər] but the final /r/ is hardly there except before a following vowel.
My dictionary gives ['hæmpʃɪə] and ['hæmpʃə], in that order.
Whoops. Having lived in Pittsburgh all my life, I naturally assumed it rhymed with Edinburgh. Turns out, not so:
Hampshire resident here, too. It’s definitely “Hampshuh” for me.
Think of it this way – “burgh” is just an alternative spelling for “borough.”
Like a Windows error message, that’s entirely correct but in the context of pronunciation not of the remotest use! 
There’s also a one-off, which we keep around just to annoy them damned furriners: “Middlesbrough”.
So from you and Colophon, combined with the Tyke evidence, I’m getting that there is a subtle distinction within the UK for “-shire”; I very definitely say “sheer”. Interesting! I’m glad someone provided a cite for my pronunciation, though, so that I know I’m not going a bit mad here…!
which is pronounced “smog city” 
Ar-can-saw with a hard R. I was born & raised in that fine state.
So, what’s a soft “R”? I’ve head of a trilled “r” but not a soft of hard one. While I’m here, I’ll mention that there’s “no noise in Illinois.” Here in SC, it is mispronounced consistently. Also, most native Missourians pronounce their state as “Missoura,” but I’ve heard some Missourians pronounce it as spelled.
And is that still with a “z” sound in the middle…?
Partly raised in Yorkshire but with an RP accent and a Lancastrian partner - I’ve never heard anyone (in real life) pronounce any county ending in shire as anything other than Whatevershuh. Number of syllables in the prefix irrelevant.
Roger that.
I don’t think that’s the case. It’s a regional split in Missouri, and I get the impression it’s perhaps 50-50, if not the “Mih-zuhr-ee” being more prevalent.
ETA:Whoa. Whole mess of Missouri/Missoura data to slog through here.