Joe Jackson
Jackson had a bit of an odd career in that it took him a while to truly catch on. He played clubs in the UK for many years before he finally had a hit with "Is She Really Going Out with Him?" in 1979. Then he changed his style to become much more jazz-influenced and he surprisingly became even more popular. His 1982 album, Night and Day, featured the wonderful earworm "Steppin' Out" and the slower-tempo "Breaking Us in Two."
Jackson's next album, 1984's Body and Soul, was also excellent, but his popularity began to wane after that. Not that Jackson seemed to care much as he was also more focused on being an artist than someone who sold lots of records. This includes making several classical records most of which have been quite well-received.
Adam Ant
Speaking of odd starts to careers, Adam Ant started as a bassist with one band and helped form his own only to have former Sex Pistols manager Malcolm McLaren take all the musicians except for Ant and make them into Bow Wow Wow. That would turn out to be McLaren's loss, however, as Ant became a pop star with double-digit top-ten hits in the UK and a couple of top-20 hits in the United States.
That was part of the issue Ant never became a global superstar. The U.S. audience mostly missed out on the cult of Ant. He had a devoted fanbase but it was small. But how can the multitude not love songs such as "Stand and Deliver," "Desperate But Not Serious," and "Strip"? In 1995, his track, "Wonderful," did peak at number 39 on the U.S. charts, but his 1980s music is so much better and completely worth one's time finding and listening to.