Inspired by the discussion of Picard’s nationality over here.
What characters have had their accents simply get up and leave or morph unrealistically over time?
The first two that come to mind for me are:
Deanna Troi, from Star Trek TNG, who started out with a multi-ethnic hodgepodge accent that seemed like it was trying to be Mumbai-by-way-of-South-London and finally wound up just being Marina Sirtis’ Londonese. Not that accents don’t change over time, but hers was inexplicable to begin with, since neither of her parents had it, and it’s clearly not the accent of any other Betazoids we’ve met.
and
Storm, from the X-Men movies, who also started with something unidentifiably foreign but in the brief period of time between X-Men and X2, dropped it in favor of sounding like … well, Halle Berry after a joint or two. “Normal,” but just kind of affectedly mystical.
Who else has examples?
Gabrielle Anwar barely kept her Irish “accent” past the Burn Notice pilot.
Kevin Costner’s Robin Hood totally sounded almost like a California guy, dude.
Daphne Moon from Frasier was supposed to be from Manchester, but sported a rather odd accent. The character was played by Jane Leeves, who really is from Manchester, and presumably knows what a Manchester accent is supposed to sound like. This was supposedly because the real Mancunian accent was mostly incomprehensible to American audiences. The producers had some with this: When the show featured Daphne’s various brothers, each of them spoke with a different accent.
Reverse this, and I have something: For all of the movies that take place in and around Pittsburgh, the only actor who kept a believable accent from that area was Michael Keaton in Gung Ho.
Of course, it was easy for him because he grew up there.
But they keep the “Shall we shoot them?” line from the pilot in the opening credits so that we didn’t forget she’s Irish. 
Besides, I think the conceit is that she covers the accent on purpose to fit in.