You need to find something that only that can do, and then the military will fund it. Like how they were ready to fund a moonbase in order to have an unreachable nuclear retaliation site until they found out you can do that more cheaply with nuclear submarines.
The non-gargantuan atomic-powered airplane gets funded that way to start with. The military is initially looking for a bomber, but they end up with an ALBM carrier (that's basically what happened IRL, except they never finished it IOTL). After that, atomic engines gets used on ABM carriers, C3I aircraft for use during nuclear war, possibly airborne sensor/RADAR platforms, and eventually transports. If you're not worried about expense, atomic aircraft are just perfect as missile platforms and C3I planes, since they can stay up there for weeks at a time. These vehicles would be in the 0.5-2 million lbs. weight class, not true monsters like the CL-1201.
We then have a convenient oil crunch, government subsidies to nuclear power to deal with said oil crunch, and readily available mass-produced aircraft reactors. I'm not yet sure this can actually make atomic aircraft cost competitive, but I'm hoping it can over long haul routes. Once civilian A-planes are flying, that leads to a natural pressure to build bigger and bigger planes for economies of scale, because reactor mass scale sublinearly with reactor power. This is helped along by the military looking for cost-effective ways to do air-basing of *Peacekeepers and similar tasks, which leads them in the same direction towards Giant Superplanes. I'm not sure how big they can actually get; that's the purpose of this thread. But examples in the 20,000,000-lb. weight class were considered by NASA studies in the 60s and 70s.
So that's the progression, A (atomic planes) --> B (civilian atomic planes) --> C (gigantic atomic planes) --> D (flying atomic aircraft carriers). I'm 90% sure I can pull off A. I'm not sure if I can pull off B or C, and I'm hoping to get a handle on the plausibility of C given B in this thread.
Incidentally, you may be interested in a short story by Timothy Zahn in the collection "Time Bomb and Zahndry Others", in which he posits the idea of the USA being covered by five giant flying aircraft carriers that are constantly doing a stately cycling loop over the country, and the idea is you fly a normal passenger jet up to these carriers and then relax in hotel-like surroundings there until you reach your destination and fly the plane down to the airport.
That was based on a real-life NASA proposal circa 1979. Although I don't remember if the passengers were supposed to shift to the carrier, or if the carrier was just a tug.
I eagerly await this timeline.

Also, I need to update my own at some point...
Thanks. Sadly, it's going to be a while. Frigging grad school, getting in the way of the internet.
