Bob
Lady Mountbatten decribed Victoria's voice as having "a distinct sort of German guttural sound about it" - and then modified this comment, adding "you wouldn't say it was a German accent, but there was a slight gutturalness about it, a slight hoarseness." So I take it that it was basically British with a German tinge to a few consonants, maybe - like the previous generation? (Edward VII and his sisters were often described as having this accent).
Lili Dehn of course commented that Alix spoke with an English accent, but I take that with slight caution now for two reasons -
firstly, I am not sure Lili could really speak about this with any degree of certainty since she wasn't British; and secondly, I feel she was slightly defensive too, and wanting to defend A. against charges of being a German spy, so an accent which came between the two would sound British to her.
On the subject of accents generally, it's worth recalling that the British community in Petersburg acquired a distinctive adge to their voices too: - James Wishaw, whose daughter Stella married one of the Meyendorffs (spelling?!) and thus entered Nicholas's court and the Russian aristocracy, recalled in his memoirs that he sent her "home" to boarding school as a child to stop her acquiring that "sing-song" accent that he considered characteristic of the Anglo-Russian community. It might be fair to assume that the Grand Duchesses had a similar accent?
Janet