For fire, in the late 19th century, there was such a thing as a telegraph fire alarm system. While invented in the 1850s, they really started getting introduced to cities in bulk by the 1860s by the Gamewell Fire Alarm Telegraph Company; here's a picture of a very early one in San Francisco. (There were police telegraph boxes too, but they tended to be more for police rather than public use, although citizens considered “reliable” could get a key.) While telephones started getting introduced not long after (the 1870s) they weren't necessarily common and widespread, and even decades into the the 20th century people didn't always know how to use them and you'd have them running or driving to a station.

In the early days of telephones, for using a telephone to call anyone, including a police or fire department of some sort, you'd dial 0 for an operator. They would then connect you. (This wouldn't always work well. The origins of the UK's 999 system was a fire in 1935, where a someone tried to call fire service but was stuck in a queue, and five people died; they wrote an angry letter to The Times leading to a government investigation and the eventual establishment of the "first universal emergency service", even though it didn't get applied across the entire country until the 1970s.)

Automatic switching (which was invented quite early, but only became common later) led to people being able to call phone, fire, or ambulance services directly, but operator calls were still around for a while. A 1948-1949 book from Ithica states directly:

Call or dial the operator and say "I want to report a fire"; "I want a policeman"; "I want a state trooper"; "I want the sheriff"; (If you cannot remain at the telephone tell the operator where help is needed).

Custom numbers had already been introduced in some locations by then and only became more common from there, but they were not standardized. The MGM movie from 1950 titled Dial 1119 -- about a man who escapes a mental institution, murders a bus driver, and takes six hostages -- gives only one of the many numbers possible for calling police.

This became the status quo all the way into the 1960s. Where the problems really became apparent was the Kitty Genovese case. It is quite likely you've heard of this case somewhere -- it's the one where, allegedly, 37 people were witnesses to a stabbing and nobody called the police -- but I want to slow down and take the case carefully, because that's not actually what happened, and the role of technology (and its failure) is important.

...

In Queens, during early hours of the morning on March 13th, 1964, Kitty Genovese was stabbed in the back four times by Windston Mosley in New York. The Marbury Apartments were nearby; one man, Joseph Fink, saw the attack happen, and went to sleep. The attacker was scared off by someone shouting out the window, although nobody came out to help Kitty.

Twenty minutes later there was a second attack, witnessed by Carl Ross. Carl called his girlfriend, and was told not to get involved; he called a neighbor (Sophie Farrar), who called the police and rushed over to Kitty while she was still conscious to wait for emergency services to arrive.

The story was later told in the New York Times as 37 Who Saw Murder Didn't Call the Police; Apathy at Stabbing of Queens Woman Shocks Inspector and not only shocked the world, but became a standard psychology textbook example of the bystander effect (the larger a crowd, the less likely an individual might help).

Lights went out. The killer returned to Miss Genovese, now trying to make her way around the side of the building by the parking lot to get to her apartment. The assailant stabbed her again.

“I'm dying!” she shrieked. “I'm dying!” She shrieked. “I'm dying!”

We have good enough documentation to say there were some people who were sufficiently able to witness events:

“We went to the window to see what was happening,” he said, “but the light from our bedroom made it difficult to see the street.” The wife, still apprehensive, added: “I put out the light and we were able to see better.”

Asked why they hadn't called the police, she shrugged and replied: “I don't know.”

However, it is not the case that nobody prior to Ms. Farrar called police. We now have a fair number of interviews indicating a variety of people tried, but got no answer. This is not at all shocking: the phone department had one line, and technology was such that just like in the fire of 1935, it was quite possible for simultaneous calls to be missed. It's also quite possible some people misdialed -- see the lack of standardization in numbers between regions as people struggle to remember just which number is police at 3 AM. (I also haven't seen anyone bring up if the chance the number was briefly unmanned; while possible, note this level of failure was not necessary during this era for calls to be missed.)

The case was shocking enough that some people close to the case were part of the national push to have a standardized emergency services number. I think it's an overreach to say the Genovese case is what caused 911 (unlike the 1935 fire and 999), but it does serve as a good example why it was needed.

President Johnson in 1965 appointed a President's Commission on Law Enforcement and Administration of Justice which in 1967 produced an influential document which included the recommendation that police establish a "single, uniform police telephone number" in addition to clearing more radio frequencies for police use. This eventually led to the 911 service, although it wasn't as rapid an adoption as you might think; fire departments were first reticent to have a secondary person who would transfer them the call and adopted a "wait and see" stance (they wanted to be called directly in the case of a fire; there was a proposal in the 1950s for a standardized fire call line, but that was in order to direct calls to the fire department, not a dispatcher). Eventually all services got on board but it took a while to get 911 to the rural areas, and 911 didn't became a completely national service until the 1980s (or in a few places, even as late as the 1990s).