Who is your Dad, Liam Neeson, aka James?
Your Dad and his team of great scientists are like, sprung, out of nowhere. NOTHING ever mention where they were from before they are at Project Purity, the Jefferson memorial, and associated with Brotherhood of Steel.
This is important because they are, and should be, a group of influential people. The ones who research how to transform irradiated water into drinkable in large quantity. An nothing else ever known before their work.
Vault? nothing ever said. BOS? While logical, but no, nothing ever support that. Based on Dr Li's fallout 4 address at Commonwealth, we can do a reverse-guess and guess they were from there. BUT, this is important, no concrete data directly support that either.
EDIT: one interesting thing is the Dr Li's science team had a few coming over from Pinkerton's science team, who before founding Rivet City came from Naval Research institute. Its root said to come from prewar institute, but you can take that however you like because there's no other datum dispute that claim. Also nothing else is said about NRI.
2nd Edit: after nearly a year continuously playing capital wasteland and mojave, I tentatively come to an educated guess: James and project purity people, including Dr Li, is from Commonwealth. The central core to this guess is Dr Li and her seemingly connections to Commonwealth. The escaped android? he hid in Rivet City where Dr Li hold most important influence. The doctor who chased after him? Also visit Rivet City to ask for their help in locating that android. This alone should showcase how much street cred Dr Li has with Commonwealth people: not a stranger, more like a known character. The android become chief of security here and the doctor was allowed to visit science lab to make a pest of himself, something only acquaintances are permitted. Otherwise Li has the muscles to throw unwanted visitors off her turf at the very least.
