I played the first two games and part of the third (which explains my avatar) but I'm not exactly a fan, as I can't remember all of the details about factions, the timeline, etc.
I lasted around 30 minutes for this show because it's not that good. The problems:
The main story is easier to understand if one starts with the vault dweller, and then introduces other characters as it travels to different places. In this case, the writers attempted to introduce three main characters right away, and then compressed them into one episode. The result is underdeveloped characterization, which leads to lack of sympathy from viewers.
Next, some of the characterization looks weird, together with appearances, as if the show was written by Fallout gamers who tried to imagine themselves playing Fallout characters. When I look at the appearances of Lucy, her brother, her cousin, the council members, Maximus, Dane, and others, they look and act in a cartoonish and surreal fashion, as if one were watching cut scenes from a Sims game with a Fallout skin, with any dark humor forced, like the weird allusions to incest and the absurd grafting of a '50s TV show goody two shoes underachiever with current sensibilities.
Finally, the set appearances makes me imagine that I'm not watching a show based on video games but video games re-imagined as a show. In short, one could have imagined just making it using computer animation, and with a comic book adaptation to follow.
Given that, I gave it around 6 or 7 out of 10 due to problems with the story. It might get better with subsequent episodes, though.