Vjylku

Probably read 15-20 years ago. English Paperback.

Scientists running a deep underground Neutrino Detector [might have been cosmic rays]. It was a VERY Large tank of water with detectors built into the tank walls looking for (flashes?) as the particles interacted with the water as they passed through (I think it happened Very Rarely)

Suddenly alarms go off, the detectors are going crazy and the tank is over-pressured. A Tech who happened to be down doing maintenence at the tank, (The control room is on the surface, and the tank is like a mile underground) opens an access hatch to the tank and discovers a man in the tank who is alive.

The story is now told in alternating points of view: People in This world, and People in the world of his origin. In his world the water tank had been designed for something else. [Maybe a super computer, or Super collider or some other advanced research tech involving super conducting magnets?]

The scientist at the facility in this world wants to know, who he is, how he got in the tank, (which is now contaminated, will need to be emptied, cleaned, repair the detectors, and refilled with (heavy water? ultra pure water?) IIRC, one of the administrators was ranting and raving that the little prank someone pulled was going to cost millions to fix) end eventually, just What is he?

The other world are focused on, what happened, where is he, etc. But I think it was more focused on his family who were the protagonist of the story. (Maybe it was a private experiment who his family were running?) I seem to recall it was a female family member (wife? sister? mother?) who was very involved in the search for answers.

The crux of the story is he was not quite human. I forget the specifics. Maybe he was telepathic. He is close to human, But the difference is enough I think to convince our world, that he must be from an alternate world. And they are trying to figure out how it happened. The man has no idea what happened. One second he was in his world in the underground facility, and next he is drowning in a huge tank of water.

I also have a vague recollection of his world it might have been a matriarchy society. I do seem to recall a separation of the sexes. Not mixing unless married?

Sounds like the Neanderthal Parallax by Robert J. Sawyer to me, which is a trilogy composed of Hominids, Human, and Hybrids.
From Goodreads:

Neanderthals have developed a radically different civilization on a parallel Earth. A Neanderthal physicist, Ponter Boddit, accidentally passes from his universe into a Canadian underground research facility. Fortunately, a team of human scientists, including expert paleo-anthropologist Mary Vaughan, promptly identifies and warmly receives Ponter. Solving the language problem and much else is a mini-computer, called a Companion, implanted in the brain of every Neanderthal. A computerized guardian spirit, however, doesn't eliminate cross-cultural confusion; permanent male-female sexuality, rape, and overpopulation are all alien to Ponter. Nor can it help his housemate and fellow scientist back in his world, Adikor Huld, when the authorities charge Adikor with his murder.

The "Canadian underground research facility" mentioned in the synopsis is the Sudbury Neutrino Observatory, which, as you say, is a large underground tank of heavy water (actually, a flooded nickel mine) used to detect neutrinos coming from the sun. I believe in Ponter's world the facility is used for quantum computer research.

The narrative alternates focus between Ponter finding himself in an alien world (i.e., ours), and his own world, where homo neanderthalensis had become dominant rather than homo sapiens, where his wife suspects he had been murdered by his [male] mate, with whom he lives periodically based on the menstrual cycles of the community's women.

My answer would be "Hominids" by Robert J. Sawyer. Many elements that you cite are in that book. The guy transferred over isn't telepathic, but a neanderthal. https://www.goodreads.com/book/show/264946.Hominids

The book is the first of a trilogy.