4 of 4 added 754 characters in body

To my knowledge, only two of the candidates that failed the tests did later become astronauts and flew on missions for NASA:

  • Jim Lovell
    According to the Wikipedia article on Project Mercury:

    Navy Lt (later Capt) Jim Lovell, who was later an astronaut in the Gemini and Apollo programs, did not pass the physical tests.

    Lovell flew on 4 different missions, notably as the commander of the famous Apollo 13. He was also part of the crew for Gemini 7 (Pilot) & 12 (Command Pilot) and Apollo 8 (Command Module Pilot).

  • Pete Conrad
    Conrad, who also took part in the selection process, disagreed with the tests practiced by NASA and chose to voluntarily fail several of them.

    Unlike his fellow candidates, Conrad rebelled against the regimen. During a Rorschach inkblot test, he told the psychiatrist that one blot card revealed a sexual encounter complete with lurid detail. When shown a blank card, he turned it around, pushed it back and replied, "It's upside down".

    Then when he was asked to deliver a stool sample to the onsite lab, he placed it in a gift box and tied a red ribbon around it. Eventually, he decided that he had had enough. After dropping his full enema bag on the desk of the clinic's commanding officer, he walked out. His initial application to NASA was denied with the notation not suitable for long-duration flight.

    He later reapplied and finally joined NASA in 1962. Conrad also flew on 4 different missions : Gemini 5 (Pilot) & 11 (Command Pilot), Apollo 12 (Commander) and Skylab 2 (Commander)

Bonus:
Edward Givens, part of the finalists, did also become a NASA astronaut in 1966 but never actually flew on any mission as he unfortunately died in a car accident the following year.

Givens had been a Project Mercury finalist back in 1959, and was one of nineteen astronauts selected by NASA in April 1966 for its fifth astronaut group. After completing basic astronaut training, he was assigned to the Apollo program and briefly served on the support crew for the first manned mission after the Apollo 1 fire, Apollo 7.