By the time of their next album, of course, Jon Anderson rejoined (and Horn stayed on as producer). So is that two changes, or one and a half? 
Deep Purple: Rod Evans, Ian Gillan, David Cloverdale/Glenn Hughes, and Ian Gillan again.
Neither band, though, depended on the initial female singers. Fairport had Ian Matthews (Thompson, of course, can sing, but he didn’t do much early on), and the Airplane had Marty Balin and Paul Kantner. So it isn’t quite the same deal as AC/DC, Van Halen, et al.
But, of course, both bands did the equivalent of trading in a Ford for a Ferrari, so that they didn’t so much change singers as acquire a great singer.
Just being pedantic, since you’re talking about my all-time favorite band (Fairport).
Though not a band but a group, and not really having a lead singer, the (IMO superb) girl group Sugababes only actually has 1 Sugababe left:
Siobhán Donaghy, Mutya Buena, Keisha Buchanan
Heidi Range, Mutya Buena, Keisha Buchanan
Heidi Range, Amelle Berrabah, Keisha Buchanan
I am expecting Keisha to depart soon, leaving us instead with Ersatzbabes.
Bad Company: Paul Rodgers, Brian Howe
Anthrax: Joey Belladonna, John Bush
INXS: Michael Hutchence, J.D. Fortune
Aghora: Danishta Rivero, Dianna Serra
The Gathering: Bart Smits, Niels Duffhues, Anneke van Giersbergen
Maybe not as soulful as with MacGowan’s drunken warble, but successful enough, sure. They recorded a couple of good albums with Spider singing; Wikipedia says “Tuesday Morning” with Spider was their third top twenty hit and their top-selling single internationally, which surprised me. Most people I know preferred Shane, but they did all right with Spider.
The Moody Blues were far more successful after changing lead singers. (Well, they had several members who sang lead, but the most prominent, Justin Hayward and John Lodge, were not there from the beginning. Pre-Hayward, they had a minor hit with “Go Now,” sung by Denny Laine, later of Wings.)
The J. Geils Band tried to continue on without lead singer Peter Wolf, with keyboardist Seth Justman taking over vocal duties, but I wouldn’t exactly call it successful.
Archetypical Christian rock band Petra successfully changed lead singers (from Greg X. Volz to John Schlitt).
Manfred Mann.
Seven top 10 songs (including two at #1) with Paul Jones between January '64 and August '66 and six top 10 entries (including one at #1) with Mike D’Abo from October '66 to April '69.
The Beatles? 
See, I came here to say
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After losing Rod Stewart, the Jeff Beck Group continued with Bob Tench and had a hit with “Going Down”.
Journey started out with Greg Rolie on vocals, but didn’t get really big until Steve Perry joined.
The Grateful Dead have lost a few singers and kept on truckin’ (I’m thinking of Pigpen, Donna Godchaux and Joan Osborne more than Jerry).
Montrose didn’t quit when Sammy Hagar left, although I can’t say how that affected their success.
Lynyrd Skynyrd seem to be hanging in there without Ronnie.
Ted Nugent replaced a good singer with himself and seems not to have suffered from it (at least in terms of his success).
Spock’s Beard is still carrying on without Neal Morse although I don’t know how successful they were to begin with.
The Moody Blues
…No, Justin was the replacement.
The best-known lead singer of Kansas is Steve Walsh, but Robbie Steinhardt sang lead on some songs. Walsh left the band for awhile in the early 80s and was replaced with John Elefante during that time. I’m not sure if you’d call Elefante’s stint in Kansas “successful,” but they did release two albums with him.
But does No Doubt count? They only became famous after Gwen took over the lead vocals, rather than the OP’s examples - both were well-established acts with several recordings under their belt before they swapped singer. One could say that Gwen was one of the major contributing factors to No Doubt’s success.