Thirty-two minutes after the first plane crashed, the New York City Department of Health (DOH) activated an Incident Command Center. Ironically, the city's Office of Emergency Management was located in the World Trade Center. It was evacuated, and communications were almost immediately rendered impossible as computer and phone lines stopped functioning.

The DOH had an emergency response plan in place, and they immediately went into action. It covered several key areas: an incident command system; laboratory, surveillance/epidemiology, medical, and environmental response capability; and communication mechanisms. In the afternoon of the day of 9/11, the DOH joined forces with the American Red Cross to set up shelters, transport necessary medical supplies to lower Manhattan, set up temporary command posts for necessary agencies (police, fire, medical, press, etc.) - From Implications of the World Trade Center Attack for the Public Health and Health Care Infrastructures.

However, as is evidenced by the documentaries we all watch, the day was chaotic and confusing.

In 2002, at the FDNY and NYPD's request, McKinsey & Company (a private consulting firm) spent five months working with their personnel to develop recommendations to enhance the Fire Department's readiness and preparedness to emergencies in a post-9/11 world. Their findings answer many of your questions. The McKinsey Report states that the FDNY received assistance from fire departments in Nassau, Suffolk, Westchester County (NY) and other neighboring jurisdictions, but with limited ability to manage and coordinate efforts due to downed communications. Besides assisting with recovery operations at Ground Zero, volunteer firefighters from Long Island and Westchester manned numerous firehouses throughout the city to assist with other fire and emergency calls. Due to limited record-keeping capabilities (as noted in the report), it's hard to pinpoint exactly which resources were diverted and how, but many outside areas sent in help to FDNY firehouses.

Regarding the NYPD, the McKinsey reportstates that one of the biggest recommendations was establishing mutual aid agreements with other agencies. According to CNN: The consultants said police did a good job moving traffic, using the subway system to evacuate people from the city center and helping to set up temporary morgues. There also was little looting... But McKinsey also found problems. Too many police rushed to the scene, and there was a lack of clear command structure and roles, lack of discipline, insufficient counterterrorism training and some disregard for radio protocol, the police commissioner said.

Many sources I have found mention the fact that "NYPD resources were stretched thin" on 9/11. I admit that I am a bit of a True Crime fan, and one of the strangest stories is that of the only unsolved homicide in New York on 9/11. It is that of Henryk Siwiak in Brooklyn, a Polish immigrant whose construction shift was cut short on the morning of 9/11 during the terrorist attacks. Later that night, he scored a cleaning shift and left his house to head to the unfamiliar building. On the way, he was lost, and presumably intercepted and killed. His case has not been solved, presumably because of the lack of resources that were able to be dedicated to it on a day when so many law enforcement officers were tapped elsewhere. “Back in 2001 there was no iPhones, there was no video cameras,” recalled Detective George Harvey. “There was a few, but not like there is today in the neighborhood … But back then, 2001, we just relied on the public to come forward and give us information.”
No one did.