Continuity mistake: When Agatha is talking to Herbert at the police line, the position of her arms varies over several shots.
Continuity mistake: This show has been widely declared as canonical with the games, by the makers of both this show and the games. The game Fallout 4 establishes the date and time that the bombs drop: Saturday, October 23, 2077, 9:47 AM Eastern Time. Which means it was 6:47 AM West Coast Time. That's a pretty weird time of day to schedule a kids' party with lots of guests and a hired celebrity cowboy.
Suggested correction: They changed a lot of subtle things from canon for the show. Also, all games have got some inconsistencies between them as well. In the show, on the TV, there is a clock that says it's around 9:30 AM the first time we see it and 10:45 when they cut the cake, so it's close to the afternoon and not very early in the morning. The position of the sun confirms this.
Second Date - S1-E2
Continuity mistake: At the deli, Jane orders plain bagels with cream cheese. When she gets home, she has everything bagels with lox.
Episode #1.5 - S1-E5
Continuity mistake: Toward the end of the episode, Pearce is cornered on the roof as he stands near the waiting helicopter. He puts his rifle down at his feet but never moves. When he lies flat as ordered, his rifle is now well behind him as he is now much further from the ledge.