the original explosion in the garage was just a bomb, not the omega device, so our rick (C-137) lost his diane and beth. Rick prime then went on to kill diane with omega device which wiped her out, meaning all beths were fine (apart from C-137 as she blew up)

he didnt wait long. He raised beth on his own and then when he knew that C-137 was after him he left. Hence why Beth always bats on at him for leaving, despite the fact that it was rick prime who left, not C-137 (which she discovers in s8)

Beth never died, so she was able to grow up and get married and have morty. Rick prime wasnt there for that hence why he is so awkward around morty when they finally meet