The change in the vowel phoneme used in "God" certainly occured in American, not British, accents. One piece of evidence indicating this is that in a usual American English accent, the vowel of "God" has merged with the vowel of "father," while this is not so for a usual British English accent. The spelling provides a clue that the distinction in British English is due to retention of an original distinction rather than due to an innovative split.
Wikipedia, which is not such a reliable source but which is the best I have right now, says the change happened "from the 18th century on."
The general development of the "short o" sound goes like this:
- something like [o] in Old English, contrasting with long [oː]
- in Middle English, it's either [o] or maybe [ɔ] later on, contrasting with both long [oː] (from Old English [oː]) and long [ɔː] (from Old English [ɑː])
- in Early Modern English, it's [ɒ], or perhaps [ɔ] early on, contrasting with both long [ɔː] or [ɒː] (from monophthongization of the Middle-English diphthong [au]) and long [aː] or [ɑː] (from lengthening of Middle-English [a] in certain environments)
- in modern English, it is variously [ɒ], [ɔ], [ɑ] or [a] depending on accent, and may or may not be merged with one or both of the descendants of Early Modern English [ɔː] and [ɑː].
Specifically:
In a usual accent from the south of England, the vowel in "God" currently tends to be somewhere around [ɔ], although often transcribed /ɒ/ due to tradition. It contrasts with both [ɑː], in "father," and [oː], in "laud."
In a usual Scottish English accent (not the same thing as Scots or Scottish Gaelic), the vowel in "God" is traditionally transcribed as /ɔ/, and it is merged with the vowel of "laud" but not with the /ɑ/ of "father."
In a usual "general" American English accent, the vowel in "God" is [ɑ], merged with the "father" vowel; in American English affected by the "Northern Cities Vowel Shift" it has been fronted to more or less [a]. In accents with the caught-cot merger "laud" has this same vowel; in accents without the merger "laud" has a distinct vowel usually transcribed /ɔ/, but often phonetically closer to something like [ɒ].
American English is not really so much more conservative than British English. Other innnovative phonological/phonetic elements that are more common in North American English than in British English are:
- neutralization of tenseness distinctions for vowels before /r/: the merger of "merry", "marry" and "Mary""merry", "marry" and "Mary", the merger of "forest" with either "fore-est" or "far-est," the merger of "serious" and "Sirius," the merger of "hurry" and "furry"
- yod-dropping (pronouncing "news" as "nooze" instead of "nyooze")
- t-lenition (voicing /t/ when it comes between vowels and the second vowel is unstressed, as in "water," or dropping /t/ when it comes after /n/ and before an unstressed vowel, as in "winter")
These are not universally present in American English accents, and yod-dropping and t-lenition exist in some British English accents as well, but these innovations are certainly more common in North American English than in British English.