I was 13 when Fallout 1 came out although I played both of them I was far more interested in Diablo II and Baldur's Gate back then so I never encountered them in the Wastes but seeing as both had Time-traveling elements and the current scinetific thought is that Time-travel creates an alternative reality whereas the actions of the reality created by the time travel is unrelated to the reality from where the time-slip came is it possible that Dr. Who created the events of the 50s culture progressing and the lack of development of the transistors and is it possible that during the years following the Great War the the star Fleet pod exited a time-slip and fell to the West Coast.
This most certainly doesn't explain the changes in science between our world and Fallout but what if Dr. Who changed Science through his manipulation of our reality or caused catastrophic damage to the Fallout Universe, whereas he completely changed how nature and science interact through the use of his reality-bending time machine and the development of the modern electrical and technological world by causing the world to evolve without transistors (the bases of modern electricity which are not in the Fallout world).
I realise its without a doubt not the cause of the divergence but I think it is neat to create an explanation for these two pop culture jokes and if there existence somehow changed the Fallout universe from our own.
Maybe TARDIS remained on the West Coast after Dr. Who had changed the world and only reactivated once you came into range of it, maybe it was even protected form the nuclear devastation because it doe not belong to the universe it created (which would likely fit the 50s view on alterante realities)?
I mean any possibility is likely when Black Isles left most of the Game-world to speculation and fan fiction.
Or maybe Dr. Who was only stopping by to be a developers joke.