Southern Hemisphere accents
Evidence for the date of the shift comes from the Southern Hemisphere accents, those of Australia, New Zealand, and South Africa.
In Australian English, there is generally agreement with southern British in words like path, laugh, class. But before N+consonant, as in dance, plant, most Australians use a flat A (aunt and can't are exceptions and are invariably pronounced with /a:/). Phonetically the broad A is [a:]. In Australia there is variation in the word castle, both pronunciations being heard.
South African and New Zealand accents have a similar distribution of sounds to Australian. New Zealand accents are more like to use the broad version. Someone who knows them in detail could add more here.
The only areas of North America affected at all by the broad A are parts of New England and the Canadian Maritimes. In these areas, in the context where British speech has /a:/ or /A:/, both /æ/ and /A:/ tend to fall together into a new sound between the two. This may preserve the conditions that led to the change in British English. This version of broad A is strongly regionally marked and is not considered a prestige variety in North America, and the area in which it holds sway has tended to shrink.