Joshua Wells was born around 1827 in Albemarle County, Virginia, and worked as a farmer before the war. He eventually made his way to Ashley County, Missouri, where he registered for the Union draft. He enlisted in the Union army on May 6, 1864, in Louisiana, Missouri, and he mustered in as a private in Company H of the 68th USCT Infantry Regiment on May 27 at Benton Barracks in St. Louis. His enlistment records describe him as 5 feet, 5 inches tall, with black hair, black eyes, and a black complexion. His regiment took part in the defense of Memphis, Tennessee, until February 1865. The men then served in Louisiana and Florida before participating in the Siege of Fort Blakely in April 1865. During the siege, on April 7, a musket ball hit Wells's left groin, causing a severe injury. The army discharged him for disability on June 24, 1865, at Greenville, Louisiana. 

After the war, Wells settled in Pike County, Missouri, and he married Lucinda Baker there on October 9, 1868. According to witnesses, both Wells and Lucinda had been previously married. Wells had a son named James, but it is unclear who James's mother was. Wells continued to suffer from his wartime injuries, and he applied for a pension in 1871. Four years later, the government granted him a $2 monthly pension backdated to 1865. Wells died on September 25, 1884, and Lucinda passed away around November 1894.