Actually the cosmology in "Megas-tu" wasn't based on the Big Bang theory, but on the competing Steady-State or Continuous Creation cosmological model that was still considered valid by many cosmologists at the time, most prominently Fred Hoyle.
How it all started: Edwin Hubble discovered in the 1920s that the universe was expanding, that the galaxies were moving apart from one another. A lot of cosmologists were uneasy with the idea that the universe was changing rather than existing in a steady state. Even Einstein, who had previously and independently calculated from relativistic theory that the universe must be expanding or contracting, had introduced the "cosmological constant" as a fudge factor to preserve the steady-state notion, though he retracted it once Hubble's observations were announced. The discovery that the universe was expanding led to the conclusion that there must have been a time when it was all compressed into a single point, that it had a finite age and a beginning, and that led to the Big Bang Theory.
However, not everyone accepted this. Hoyle and others proposed the
Steady State theory in 1948, suggesting that as the universe expanded, new matter was created to take the place of the old, so that the universe's matter density remained constant and there was no need to presume a finite age to the universe. There were various ideas as to where the new matter came from, but one was that it spewed into our universe through white holes connecting to other universes (i.e. what fell into black holes there came out here). Such "white holes" were one proposed explanation for quasars, so the idea that there were white holes in the centers of galaxies was not unknown in the '60s and '70s.
Today we see the Big Bang as the default theory for the origin of the universe, but that's because of all the evidence that has accumulated to support it over the past few decades. In the early '70s, some of that evidence had begun to emerge (mainly the cosmic background radiation, the afterglow of a vastly hotter early universe), but most of the evidence we have today hadn't been found yet, and there were still a lot of supporters for Steady State as an alternative model.
So the writer of "Megas-tu," Larry Brody, either was a believer in Steady State or decided it offered better story possibilities than the Big Bang model. Either way, he took a gamble on one of two competing theories and just happened to back the wrong horse. Which is an occupational hazard of science fiction.