In the Doctor Who episode "Last of the Time Lords," Jack unwittingly reveals himself as the future Face of Boe, but it is not said when this transition occurs. Writer Russel T. Davies previously halted production of comic books or spin-off novels that definitively state the truth or falsehood of the theory, but on March 30, 2020 on Twitter, Davies confirmed that the Face of Boe and Jack Harkness are indeed one and the same.
In series 3, episode 13, Jack evolved into a different kind of being.
Jack's ponderance is a conjecture of what he'd look like if he reached around the age of 1,000,000. At the time of the episode, he was about 2,158 years old due to periods of being buried alive in "Exit Wounds" and then frozen an additional 107 years. As of today, he'd be about 2,177 give or take some overlaps in his timeline. Davies said that it would be entirely realistic for Harkness to look like Boe in 5 billion years after his nose and ears keep growing due to age. Enormous alterations are possible in all that time, and for the immortal traveler, this is how those changes manifest. As to why the Face of Boe has a British accent where Jack Harkness's is American remains a mystery, as does how Jack lost the rest of his body.
With Jack's hypothesis of evolution happening around 1,000,000 years and viewers first seeing the "Fugitive" that "nanogenes are always after me," so it seems very likely that this attack is the start of his transformation (early in the third decade of the 21st century).
Taking into consideration all the possibilities for Jack's mutation, the most likely reason for his final form is tied to his ultimate resting place. The exhaust in New New York developed sometime between 4,999,999,935 and 5,000,000,053. Jack's head survived the nanogenes, growing several times its original size sometime in those 65 years, transforming Doctor Who's beloved Jack into the Face of Boe.