I don't know anything about the change in pronunciation of this particular word, so this is just a partial answer.

The more general sound change this is a part of is shortening of /uː/ (from Middle English /oː/) to /ʊ/. This is a sporadic sound change that occurred in RP as well (in other words) and is one of the major sources of /ʊ/ in Modern English. For example, it occurs in foot, soot, good and almost all -ook words.

This sound change shows up in a number of words, but not in tooth, in the 1644 text The English primrose, by Richard Hodges, which is written with diacritics indicating the pronunciation of the words. Hodges writes hood and good as<hꝏ̯d>and <gꝏ̯d>, with a vowel that he describes as a short version of the vowel in words like tooth (which Hodges writes as <tꝏth>). Hodges also has short <ꝏ̯> in <hꝏ̯p>, unlike in modern RP. Hodges is cited in The Cambridge History of the English Language, by Roger Lass (2000), which is cited by the Wiktionary article "Phonological history of English close back vowels: FOOT–STRUT split". Based on the way Hodges and Wallis (another source on early pronunciation) describe the sound of the shortened vowel in words like good, Lass says that it was most likely a short vowel with the quality [u] at this point in time, rather than [ʊ] (Lass 90).

Even though Hodges doesn't provide direct evidence for the pronunciation of tooth with a shortened vowel existing at this time, the presence of the shortening sound change in other words makes it seem plausible that tooth might have had a shortened vowel for people other than Hodges in 1644. However, it's hard to estimate how recently the sound change might have applied, since there isn't anything I know of that would prevent it from occurring more recently.

Sporadic shortening in Early Modern English is also attested in the RP pronunciations of some words that had Middle English /ɛː/, which regularly developed to modern RP /iː/ (as in heath, beneath, mead, wheat), but in some words such as death, breath, dead, thread, threat developed instead to short /ɛ/.