I found this interesting Chess.com article which goes into the question in more depth. It gives short biographies of Paul Nemenyi, Regina Wender Fischer and Hans-Gerhardt Fischer along with some details for Joan Fischer, Bobby's older sister, and Peter Nemenyi, his likely half-brother.
Here is how the article describes how Nemenyi and Regina Fischer met:
In 1942, Dr. Paul Nemenyi, age 47, met Regina Wender Fischer (1913-1997), age 29, in Denver, according to FBI files. Regina was taking classes at the University of Denver while working at a company that made chicken incubators. She was a mother of a 5-year-old girl, Joan Fischer. Her husband, Hans-Gerhardt Fischer, was in Santiago, Chile. He was barred by immigration authorities from entering the United States.
In the summer of 1942, Dr. Paul Nemenyi and Regina Fischer were romantically linked, probably had an affair, and he may have fathered Bobby Fischer.
and Regina and Hans-Gerhardt Fischer's early life together:
Hans and Regina moved to Moscow in 1933 and were married in Moscow on November 4, 1933. She was 20, he was 25. He was a biophysicist working at the Moscow Brain Institute. She was studying medicine at the First Moscow Medical Institute and he was studying biophysics.
Their first born child, Joan Fischer, was born in Moscow in 1937. She died in 1998.
On July 29, 1938, Regina went to renew her passport at the American embassy. She told a staff member that she had separated from her husband. At about this time, Hans-Gerhardt had left or was sent to operated on the Republican side in the Spanish civil war to fight the Fascists.
In late 1938, Regina and Joan departed for France and met up with Hans-Gerhardt in Paris. It's possible they also went to Austria.
On January 23, 1939, Regina Fischer left France and headed to the United States with her daughter, Joan. ... She caught one of the last ships leaving France for America. It is not clear if Hans-Gerhardt was on this ship (he did not have a U.S. passport, but had a Spanish passport). The FBI does say that Hans-Gerhardt Fischer never entered the United States from any ship (was he a suspected Soviet spy?).
The FBI file says that Hans Gerhardt Fischer lived for a time in Port San Antonio and Santiago, Chile where he sold and installed fluorescent lights and worked as a photographer. He landed in Chile on January 4, 1940.
It goes on:
Regina later divorced Hans-Gerhardt on September 14, 1945 on the grounds of willful neglect to provide for her and her two children. At the time of divorce, Regina was living in Moscow, Idaho. He never lived with her in the United States. He remained a lifelong German citizen.
So, unless she visited her husband in Chile in summer 1942 it is impossible for him to be the father.
Dr. Nemenyi took a deep interest in Bobby Fischer and even paid child support to Regina. At one time, in 1947, when Bobby was 3, he complained to a social worker about the way Regina was raising Bobby. He told the caseworker that Regina was mentally upset and Bobby was an upset child.
In 1949, Dr. Paul Nemenyi went to a social worker again, complaining that his son was not being brought up in desirable circumstances, due to the instability of Regina.
So, Nemenyi acts like Bobby's father. Then:
Dr. Paul Nemenyi died of a heart attack on March 1, 1952, at the age of 56. He had just stopped at a dance at the International Student House in Washington, DC. There, he dropped dead of a heart attack. He was living in Washington, DC, and working at the U.S. Naval Research Lab. He was survived by his son, Peter , a civil-rights activist. Peter wrote that his father, Paul, was the father of Bobby Fischer.
At the time of his death, Paul was paying for 8-year-old Bobby’s education and sending $20 a week to Regina.
Then the question of what to tell the young Bobby Fischer?
Regina did not want to tell Bobby of Paul Nemenyi’s death and was hoping that Peter Nemenyi would do it. He was not comfortable with that, so he consulted a family doctor for advice. He wrote to his family doctor, “I take it you know that Paul was Bobby Fischer’s father. The matter is further complicated by the false pretenses about Bobby’s identity and the parents’ difference of opinion over the question.” Peter felt he was not qualified to tell Bobby about Paul’s death since Peter had met Bobby only a few times.
The full article is a fascinating read and lays out strong circumstantial evidence that Nemenyi was Bobby's biological father.