The stories that the Notion Club members see are events at different times in their own universe.
In-universe, The Notion Club met in Oxford in the 1980s and then their minutes landed in a sack of wastepaper in the basement, until they were uncovered and published in the 2010s.
The Club members witness a number of different events across different points of time. Some of these, like King Sheave and Saint Brendan, have historical basis. Others, like the Fall of Numenor or Ælfwine of England sailing to Tol Eressea, are completely made up by Tolkien.
These are the same set of stories that Tolkien was using for his previous unfinished time travel book, The Lost Road. (NCP is essentially a reboot of TLR)
The real world components are meant to be our world. The parts taken from Tolkien's legendarium are meant to be from his legendarium. None of the stories are ever implied to be a parallel universe, and there are explicit references throughout to the theme of things being at different points of time. The implication is that these are all in the same universe.
This is the straightforward way to read the text (or at least as much as there is a straightforward way to read NCP).
That said, the specific things that you refer to in your question, such as "Reunited Kingdom of Gondor and Arnor" and the red book of westmarch, would not have existed in the Notion Club Papers, because at the time that Tolkien wrote NCP he hadn't yet invented them.
