I suppose this depends on how you define terms, but it seems to me that a few people could be said to have had two completely separate careers in show business.
In my opinion, the prime example of this is George Burns. The main part of his first career was his radio and tv shows with Gracie. This first career was a partnership. Then after she died, he reinvented himself, and had a whole second career by himself, as a “two-pronged” comedian who wrote several books and had major roles in several movies.
Another sort of second career is exemplified by Rodney Dangerfield. He had a first career as a comedian from 1940 to 1949 under the name Jack Roy. Then he did other things like selling aluminum siding for a while. He then went back into comedy from 1962 till his death in 2004 using the name Rodney Dangerfield and using the “no respect” persona we now know him for:
Cher is a somewhat unusual case of a crossover. She became famous for singing and for her sketch comedy TV series. Many other celebrities have crossed over into acting from a similar background. But they generally do so in lightweight comedies. Cher jumped into the deep end with dramatic roles in Come Back to the Five and Dime, Jimmy Dean, Jimmy Dean, Silkwood, Mask, and Suspect.
Stand-up comedian (and/or comedy writer), then actor, is a pretty common one. This would also include Woody Allen, Bill Cosby, Tim Allen, and plenty more.
Conan O’Brien went from TV writer to talk show host.
Jimmy Buffett - first an unbelievably popular singer/songwriter/musician, then one of the less than a dozen authors to put books at the top of the NYT Best-Seller list in both Fiction and Non-Fiction, then as a ridiculously rich businessman.
Plus he’s a really nice guy.
eta - He’s also acted a little and written a musical (it tanked.)
Other entertainers who achieved success in “show business for ugly people” include Ronald Reagan, Fred Thompson, and Al Franken. Too soon to tell for Donald Trump.
To be closer to the spirit of the OP, I’ll mention US actor/German singer David Hasselhoff.
Are there any people who simultaneously have two unrelated show business careers? For example, Charli XCX writes and performs Top 40 pop songs for herself, and writes Top 40 pop songs for other people. But is there anyone who writes and performs pop songs for themselves while writing Icelandic death metal for others? Or is there a writer/director who only writes lowbrow comedies for others to direct, and only directs action movies written by others?
Like, take Clint Eastwood: he’s a big enough movie star that he starts directing films where he’s top-billed – and ultimately wins Oscar after Oscar as a director, but only with Oscar-nominated acting performances. So would that be separate?
(Of course, he also earned nominations and awards directing films without doing any of the acting – FLAGS OF OUR FATHERS, LETTERS FROM IWO JIMA, INVICTUS, BIRD, MYSTIC RIVER, AMERICAN SNIPER – so figure he eventually earned it; whereas, say, Warren Beatty branched out from acting to win an Oscar for directing himself, but he’s really only ever done projects where folks keep watching him act before asking was that a well-written line by Beatty, or did Beatty just deliver it well? Does that count as good acting by Beatty, or getting a good performance out of Beatty?)