A statement didn't give any further details about the 29-year-old's condition.
It comes after Nelson recently missed the final of the girl group's BBC One talent show, and their hosting of the MTV European Music Awards.
Little Mix's latest album entered the UK chart at number two on Friday.
All six of the studio albums they have released since forming on The X Factor in 2011 have gone into the top five, and they have had four UK number one singles.