I have a web site which includes a page that runs through the history of typeface designs. Three supplementary pages deal with special topics. One of them focuses on the typeface Cushing, which seems to be the one used for legibility in some old Bibles.
However, to narrow down the identification more positively, because one similar typeface also used in some Bibles appears to be the Mediaeval Egyptian from Bauer, I tried making use of the fact that this typeface was used in some pronouncing Bibles.
Due to the vagaries of English spelling, if one wishes to mark up English text with diacritics that indicate the pronounciation, without altering the spelling, one needs quite a few diacritical marks. (Some dictionaries also did it this way, instead of including a pronounciation in parentheses after the word; indeed, one pronouncing Bible noted that it used the system from Webster's dictionary with permission.) And, indeed, looking at the keys to the systems used, they used diacritical marks that I couldn't find among the characters in, say, the 1905 Linotype Specimen Book. Not even those used in an 1897 Bible.
So I may be missing a part of the story.
I can only guess, as my knowledge of history in English typography is scarce.
I guess that they reused diacritics from other writing systems like French, Latin. Latin orthography/typography changed a lot over the centuries. In scientific botany ligatures and diacritics are still used for scientific names. This is against the rules for nomenclature (ICBN), which only allows the 26 letters.
A transcription from one writing system to an other uses the phonetics and writing system of the target language. E.g. in English it's 'Yiddish' and in German 'Jiddisch'. Modern dictionaries like Beinfield, Bochner, Comprehensive Yiddish-English Dictionary, 2002, uses pronounciation hints for words of Hebrew origin in a special way like e.g. [HAKhNO'EDIK], thus not IPA.
As an opposite example I have a Czech grammar printed 1902, printed in German as the main language, which uses IPA.
I have only one Bible in English, 'Appointed to read in churches', printed 1916 in London at Cambridge University Press. It's a pocket version (8 x 13 cm, 1016 pages) without diacritics, partly using a very tiny Italic (3 pt?), main text 5 pt(?).
The specimen of Linotype looks more like IPA. U+0324 COMBINING DIAERESIS BELOW is defined in Unicode and below o it's AFAIK part of IPA. IPA was introduced 1886, but not accepted as we can see.
Found something similar, maybe partly inspired by the British variant, in a German-English dictionary by Langenscheidt, 1902, used by my mother (born 1931) in school.
They called it Toussant-Langenscheidt system claiming that's their own invention.