Daphne's least favorite brother, Simon (played by Australian Anthony LaPaglia) has a terribly unconvincing accent. Mostly played for laughs, though.
See also any guest actor (not-British) playing Daphne's random boyfriends in earlier seasons. See British Accents above.
Jane Leeves is British but cannot do a Manchester accent for toffee. God help the woman if she ever needed directions to the Trafford Centre. Word of God is that they wanted Daphne's accent to be working class and understandable to Americans so Jane Leeves adopted that particular accent. 
Gil Chesterton is played by an American-born English actor and displays many symptoms of I Am Very British. In one of the first episodes, Frasier comments on his "phony accent."
Kirstie Alley is the only actor from Cheers to not reprise her role on Frasier, reportedly because Scientology does not believe in psychiatry, and Frasier is a psychiatrist. In the later episodes of Cheers, around the time Lilith and Frasier were going through their separation, Frasier and Rebecca nearly ended up in bed together. Wasting a perfectly good UST plot...
Kirstie Alley claimed in one interview that her being a Scientologist had nothing to do with her never appearing on Frasier, and at least one of the show's writers (Ken Levine) said in a radio interview that they never considered having Kirstie Alley on, mainly because they weren't sure how to write her in, as Rebecca and Frasier hardly interacted on Cheers. Alley also said once she auditioned to play one of Frasier's girlfriends of the week, but was turned down, because the producers thought that would be kind of weird.
Lisa Kudrow was originally cast as Roz, however after the first few days of filming the producers decided that her quirky humor didn't fit the part and they hired Peri Gilpin instead (according to Kudrow, Gilpin was their first choice, but switched her for Kudrow for some reason, then simply restored order by bringing Gilpin back.) Plus, imagine how this would have affected Friends if Kudrow had remained on Frasier.
Frasier's brother Niles wasn't in the original concept and hadn't been mentioned on Cheers. The inspiration for the character came after the producers saw a headshot of David Hyde Pierce and noted his brotherly resemblance to Kelsey Grammer.