The couple, both 28, wrote on Instagram that the child was born on Saturday.
They shared black and white images of the child’s head and feet, but did not reveal any name or gender.
Edwards had announced that she was pregnant in May, writing that she was “so happy to be on this wild trip with my soul mate”.
She has received kudos from many of her 12.7 million Instagram followers, including “Love Island” presenter Laura Whitmore and singer Stacey Solomon.
Meanwhile, Oxlade-Chamberlain was congratulated by his English teammates Mason Mount, Trent Alexander-Arnold and Jordan Henderson.
Little Mix rose to fame in 2011 after winning the UK TV show “The X Factor”.
They became one of Britain’s biggest girl groups, with five UK number one singles and one number one album.
The rest of the group, Edwards, Leigh-Anne Pinnock and Jade Thirlwall, have said they intend to continue performing and touring after Nelson leaves.