In his entry for here, Harold Wentworth, American Dialect Dictionary (1944) mentions several word combinations that yield a ch sound at the junction of the preceding word and the word here.
In the International Phonetic Association alphabet, which Wentworth uses for word pronunciations, the typography for the ch sound is tʃ; the ʃ character in isolation represents the sh sound, and the j character (which appears in combination with t as tj, as an alternative to tʃ in some pronunciations) represents the y sound. The modern IPA alphabet's typography seems to have changed slightly from the 1944 form that Wentworth uses, but most of the characters that I use below are true to the form he uses. (My thanks to tchrist for pointing me to an easy-to-use online keyboard of current IPA characters.) The relevant dates and regions given in American Dialect Dictionary for instances of ch sounds used in the context of t + here are as follows:
1937–40 w[estern] N[orth] C[arolina] |raɪˈtʃɪɾ̩| = right here. ... 1939 pseudo-Negro 'Wait a minute here' (|weˈt ə mɪˈnɪtʃiˈɾ̩|. Radio, Amos & A[ndy] 1939 W[est] V[irgini]a right here |raɪ'tji'r| |raɪ'tʃir| 1939–44 n[orthern] W[est] V[irgini]a In such phrases as get here, out here, right here, wait here, out here, |ʃ| or |ʃj| very commonly replaces |h|. 'I believe I'll eat here ðis evenin' ' |ə bli'v əl i'tʃi'r ðɪs i'vnɪn|. 'Right here' |raɪ'tʃi'r|. 'Wait here for us' |we'tʃi'ɾ̩ fɾ̩ ʌ's|. 'I'll get here by 4 o'clock' |ɑl gɛtʃiɾ̩| .. 'Out here' |ɑʊtʃiɾ̩|. 'Was you talkin' about here?' |əb[ɑʊ]tʃiɾ̩|. 1940 n[orthern N[orth] C[arolina] [sɪtʃiɾ̩] Sit here.
It thus appears that in the 1930s and 1940s, pronouncing "here," when it followed a word that ended in t, as "cheer" was common across a region encompassing much of West Virginia, western Virginia, western and northern North Carolina, and eastern Tennessee, and perhaps parts of eastern Kentucky as well—that is to say, much of southern Appalachia.
Wentworth was a professor in the English department at the University of West Virginia, so he undoubtedly was quite familiar with this pronunciation.