In the third Men in Black movie agent J finds himself in an altered timeline, one in which agent K was killed by Borris. What I am wondering is why wasn’t J himself also dead in this timeline? We know he was present at the final confrontation, and by the end of the trip to the past Borris attempted to kill him along with his father.
This is because in the original Timeline we can assume events played out much the same in the Restored Timeline before J went back to the Future.
In the final part we see Borris kill J's father who was protecting K from Borris's Attack. after this we see a young J run up looking for his father before k neuralyzed young J. from Wikipedia
1969 Boris attacks K on a beach, but the colonel saves K by sacrificing himself. K kills Boris instead of arresting him as he originally did. The colonel's young son arrives and inquires about his father, but rather than tell him the truth, K neuralyzes him and tells him only that his father is a hero.
if what Griffin said is true and that things would play out as they were meant too after Borris lost his arm then everything from after that was the same as the Original Timeine with the exception being K Killing Borris instead of Capturing him. with the final scene of K meeting J implying that in the original Timeline and even from back in the first film, K already knew J.
But as to why J even woke up after the Altered Timeline is because he wasn't a threat to Borris and thus not killed. Neither Borris knew that J would time jump too and stop Future Borris (which may explain K's decision to kill Past Borris, to stop a situation where Borris has foresight of J's involvement).
As we see Future Borris go to shoot K while on the Apollo 11 (before J Stop him) we could assume that in the Altered Timeline because K was dead and never returned there was no event in which Colonel Edwards was killed protecting him from a pursuing Borris. this could suggest that J had a father at the time the Boglodites attacked in the Altered Timeline but because he still remembered K and the Arc Net Shield and Lunamax (which was built for Borris initally) it's possible he still believed his father was dead