While the FBI and other law enforcement knew exactly when Gambino died, I wouldn't be all that sure that they knew immediately who succeed Gambino. As DC said, that scene was more Hollywood than real life.
The movie took other liberties, too. For example, Castellano didn't force Gotti to take Ralphie (Wigs) Galione on the McBratney hit. Nor did Gotti's guys kill Galione. Wigs was killed about a year after the McBratney hit over a debt he owed to a loanshark. Also, poor John Favara, the neighbor who accidentally ran over Gotti's son Frank, wasn't killed in an underpass--he was hit over the head after leaving a diner, bundled into a van, and never seen again. And it's unlikely that Da Bull had anything to do with the Favara hit.