My interpretation is that there has been some sort of organization devoted to containing SCP-style anomalies basically since people realized that they COULD contain some of them (or at least try to). These organizations of course had varying levels of success. I think that large-scale successful containment didn't start until the Enlightenment, at which point a lot of small groups started in the West, similarly to how a lot of museums started from cabinets of curiosities. There were still many smaller groups scattered throughout the world (although I could see a larger, more bureaucratic organization already in place in China), but the ones in the West were the first ones to start banding together into something resembling the Foundation as it exists today.
At first, there were a lot of national separations (the English didn't trust the French & vice-versa, for example, and the Americans were too far away from Europe for anyone to care about what happened there except the Americans, etc.), but those gradually melted away as rapid(ish) transit became more possible. The invention of the airplane helped A LOT. As it was, I think that these various proto-Foundations had different responses to their nations' governments; some were outright government agencies, some were secretive, shadowy "brotherhoods" like the ancient Greco-Roman mystery cults, some were just groups of business or aristocrats seeking to acquire these marvelous objects for their own edification (MCD, anyone?).
In any case, at some point these organizations decided that merely "offering assitance" to their fellow containment teams didn't make sense when they could be a whole organization. There was definitely a period of chaos as the various structures were integrated, including divesting themselves of their overt governmental connections, until they ended up with the Foundation as we know it. They still maintain contact with various governments, but (usually) behind the scenes and only with a very, very few number of people, and rarely the heads of state (they rotate in and out far too often to be trusted).
My feeling is that the proto-Foundation organizations started feeling each other out to unify sometime around World War I, but finally came together during or in the immediate aftermath of World War II. The American proto-Foundation had originally been a military project and the government/military was heavily involved up to that point, maintaining a security role until sometime in the late 1950s to mid 1960s. Other governments, especially ones where their proto-Foundation had never had extensive government contact, didn't even notice the difference.
And some never really broke contact at all. As I mentioned earlier, I think the China had a better organized proto-Foundation for a while before the others got their act together, mainly due to the bureaucratic structuring of Chinese government and the cultural high degree of heirarchical structure. They were one of the, if not THE, last organization to join the Foundation, and even today they maintain a degree of independence from the rest of the Foundation. They have a fairly tight communication chain with the Chinese government, who freely transfers responsibility for handling these matters off to the group that has been doing so for ages.
Anyway, that's my head-canon.