How did “Papa” become “Pope”?

Solution 1:

The Old English form was apparently pronounced pāpa, with a long back open vowel in the first syllable (in the modern representation of Old English, ā represents a vowel reconstructed with the quality [ɑː]). A regular sound change turned Old English ā [ɑː] into Middle English [ɔː], a long back mid-open rounded vowel that was generally written with the letter o. And Middle English [ɔː] regularly develops to the modern English "long o" sound, which is spelled the same way.

The long vowel in the Old English form corresponds to a long open vowel in Latin:

• Lat. pāpa, signifying properly 'spiritual father', i.e. 'pope', was borrowed by O.E. unchanged from the Latin in the form of the weak masc. pāpa, -an [...]

  The Latin title „papa“ was, as the reader is doubtless informed, at first applied to all bishops indiscriminately. And it was not until after the 5th Cent. that the Roman pontiff alone began to be addressed as ‘papa’. Consequently, O.E. pāpa, borrowed from the Lat. not until after the aforesaid period, designates only the pope.

  (The Influence of Christianity on the Vocabulary of Old English, Volume 1, by Hugh Swinton MacGillivray, p. 83)

I assume that the question "how and when did the vowel change come about?" is about the first vowel. The final -a of Old English pāpa would have turned into a schwa and then been lost. Word-final schwas were sometimes spelled with -e in Middle English, but after these schwas were lost, final -e was reinterpreted as a mark of vowel length (the so-called "silent e"). Because of this reinterpretation, final -e came to be used in the spelling of some words that hadn't originally been pronounced with a schwa sound, so there is no reliable association in either direction between the presence of a final -a in Old English and the presence of a final -e in modern English spelling. For example, Old English fola corresponds to modern English foal, while Old English dāl and rāp correspond to modern English dole and rope.