NYPD was first on the scene and arrested Frankie (Tom: "The NYC detective squad said Frankie was half dead, scared, talking out loud about how you betrayed him"). Frankie was inside the bar, not out in the street where the shootout took place. Perhaps the detectives, seeing that Frankie was ready to rat out Mr. Big, thought it would be in everyone's best interest to keep Frankie's survival secret, the better to protect him against Michael trying to silence him, and to get more info out of him.
One problem is that if everything isn't coordinated, then the Rosatos are going to be trying to figure out where Frankie is, which again would leave some sort of trail for the Corleones to follow.
The "directorial license" part is how the Senate subcommittee managed to wheedle Frankie away from NYPD. A stretch-guess: The Senate subcommittee already had hearings scheduled on organized crime, and was planning to call Michael as a witness. Roth, through Questad, had the Senate subcommittee chair appeal to NYC's mayor to have NYPD give up Frankie to the subcommittee. The promise would have been that the subcommittee had a far better chance of connecting Michael, through Frankie's testimony, to a greater range of crimes (i.e., running all the gambling in America) than were covered by NYPD's jurisdiction.
The fact that Frankie was willing to talk would necessarily bring the feds in, right? I realize they didn't have a formal witness protection program, but they did stuff like that on an ad hoc basis and the the FBI and Justice Department would have to sign off on all of it to give Frankie meaningful immunity.
