My understanding is that officially 6 people died, likely more given the tendency to undercount poor and unhoused populations. And of course, likely even more if you include people who died indirectly as a result of the economic impact (starvation from lost income, etc.).
But 6 people seems absurdly absurdly low, even as an official count. Given how narrow City streets were (and are), I would have expected more people to die from evacuation alone (trampled, crushed by someone's overloaded cart, human crushes, etc). And I know the fire began in the middle of the night in one of the most densely populated parts of one of the most densely populated cities in the world. Given the apparent chaos, I'm surprised there weren't more than 6 fraudulent claims of death (e.g., people using the fire to disappear or people collecting insurance money under false pretenses).
So what happened? What factors kept the official death count relatively low? Were their factors that also reduced the unofficial death count?
I recently wrote an essay about the Great Fire so can offer a short and simple explanation based around some, but not all of the key points.
1. The fire spread slowly since London was built of low-rise, timber-framed buildings
This is perhaps the most significant factor because it meant while the fire spread between buildings consistently and indiscriminately, it also took a long time. In fact, the fire burned for a whole five days, with Pepys even reported seeing smouldering remains several months later. That is a long time for a fire to spread through only one square mile. (By way of comparison, up to 800 square miles burned every day during the Australian bushfire of 2020.)
Why was this? Well, as you'll know if you've ever thrown a log onto a fire, dry timber catches easily, but it takes a while to catch ablaze, and even thereafter, burns slowly, never producing a particularly large flame. London at the time had all the properties of a giant log fire, with lots of thick timber but very few combustible catalysts as you would find in a modern city (e.g. petrochemicals). Many buildings also had walls of brick or stone, offering additional fire protection. (Take a look at what 17th century might have looked like in this reconstruction.)
Also, the houses of 17th century were far smaller than the tall, densely-packed buildings of today, meaning that once a house had caught, most people were no more than two or three flights of stairs away from the street. Evacuees had ample time to exit their homes in an orderly manner.
2. London had a fire brigade who helped to slow the spread
A popular myth suggests that London only implemented its first fire brigade after the Great Fire, but the reality is that London had actually established a semi-professional fire brigade after the earlier (and much deadlier) Great Fire of Southwark.
This fire brigade was limited in its effectiveness. Without modern fire-fighting equipment like hoses and extinguishers, their best tactic was to demolish buildings by pulling them down using so-called "fire hooks", and digging trenches in the fire's path. Creating physical space helped to prevent the lick of flames from catching other buildings.
Individual people also joined in the efforts to slow the spread, throwing buckets of water gathered from the nearby Thames. This image is what their combined efforts might have looked like – note the men using fire hooks on the right.
3. Evacuation channels were plentiful
Once they were out on the street, evacuees could quickly find their way to an evacuation channel and make good their escape. Perhaps the most common route was to head south to the Thames, where a conveyor belt of ferry boats was waiting to transport people over to the south bank. Many others headed east towards the Tower of London, which was mercifully built of fire-retardant stone, or west into the adjacent city of Westminster. There was no need for a stampede, for the fire burned so slowly behind them that most people never actually completed their evacuation, but instead found a good viewing spot and turned around to watch the fire burn, as can be seen in this painting by Robert Hooke.
4. The response to the fire was well-orchestrated
You are right to bring up the point of economic factors affecting people after the fire. It is true that many people lost their lives and livelihoods and became refugees. It is likely that a great number of people died as a result of their refugee status. Refugee camps popped up in the north and east of the city, while some people rebuilt makeshift homes on their original plot of land atop the smouldering remains of the city.
However, charity was also great. The king and church were eager to be seen to be helping and raised hundreds of thousands of pounds to replan and rebuild the city. Even France, with whom England was at war with at the time, sent charitable donations to help the disaffected. While rebuilding was slow, taking up to 50 years in some parts of London (owing to the fact the king chose to use the fire as an opportunity to replan London), many people were ultimately re-homed by the local authorities.
Hope that helps. There is a great and detailed historiography of the Great Fire, with an excellent primary source in the form of The Diary of Samuel Pepys, if you were inclined to read in more detail.
