How Much Money and Will do you Have?
There are a vast number of variables in this question, but the simplest explanation is: As fast as you can spend the money. There must be actual valuable resources to be found in the area if it will be sustainable. People must WANT to live there or have no alternative. It must be able to sustain agriculture, and your society must have the technology to master the environment. There can't be effective and determined native resistance or such efforts are pointless.
Sebastos, or Caesarea by the Sea, was built by king Herod and the Romans to secure the region economically and politically. The Harbor, and most of the city, was built in relatively undeveloped territory with no existing harbor, and were built in about 12 years. To this day it is considered an engineering feat. To built a colony from scratch to operational city, this is about the minimum if you have essentially unlimited resources, manpower, a favorable location and will to do so.
With less resources, building in an organic way, you'll have something more like the Viking colonization of Greenland. With the compelling demand for walrus ivory, the Norse moved into Greenland, at the peak having about 4000 inhabitants. While the timeline to "successful colony" is a little awkward to nail down, it was a growing thriving place within a century. But this illustrates the problem with establishing colonies. Within 400 years, the colonies were abandoned. Why? The central compelling trade item (ivory) dried up, but a successful colony would have survived at least at subsistence level. It has been suggested the Norse failed to adapt to local conditions and disregarded the native traditions that could have helped them. This theory is in dispute. The little ice age likely contributed to the Norse agriculture being unsustainable, as may the black death. Some people even suggested it was simply a place people didn't want to live, so they left.
So at least a decade for a colony in a favorable spot with unlimited resources. Anything less, and you'll need will, a core resource to make it profitable, and sustained conditions to allow the colony to survive any of the frequent downturns that will strike such places on the edge of the civilized world.