As for BoS, read:
http://fallout.wikia.com/wiki/Brotherhood_of_Steel#History
As for Europe:
Fallout intro:
In 2077, the storm of world war had come again. In two brief hours, most of the planet was reduced to cinders. And from the ashes of nuclear devastation, a new civilization would struggle to arise.
Fallout 2 intro:
The earth was nearly wiped clean of life. A great cleansing, an atomic spark struck by human hands, quickly raged out of control. Spears of nuclear fire rained from the skies. Continents were swallowed in flames and fell beneath the boiling oceans. Humanity was almost extinguished, their spirits becoming part of the background radiation that blanketed the earth.
Jesse Heinig, one of the FO1 designers:
One of the recurring themes of Fallout is that life will find a way to continue, albeit often under great struggles and with violence and suffering. It's not unreasonable, given this notion, to presume that U.S. remnant forces remain in parts of China, just as remnant Chinese elements are in the U.S.; and that other countries are similarly ravaged and war-torn, with survivors crawling out of the rubble. If Australia was untouched by the war, for instance, then presumably after 200 years they would have projected their powerful industrial presence and comparatively high population all around the globe to take control of any remaining resources, and the Enclave would find itself locked in a war with the Aussies. It's likely that some underpopulated parts of the third world escaped the full brunt of nuclear devastation, but since these would have been low-population unindustrialized areas anyway, they are not exactly in a position to take advantage of their "good fortune," such as it is. (I don't imagine that many nukes were wasted on the Sahara.)
The very first Fallout 1 timeline by Scott Campbell and Brian Freyermuth (from which all other Fallout timelines in existence are derived):
Other countries, seeing the US's missiles on their way, fire their warheads as well. What ensues is two hours of nuclear bombardment upon the earth's surface.
Emil Pagliarulo, FO3 lead designer:
Tenpenny Tower was slightly inspired by Fiddler's Green, the skyscraqer in George Romero's Land of the Dead. But it was also an opportunity to introduce another character from outside the U.S. Allistair Tenpenny came to the Capital Wasteland from Great Britain to seek his fortune, so that alone tells you that the U.K. was also hit in the war. And if he came to U.S. to succeed, that says a lot about how screwed up Europe must be. So we just allude, a little bit, to the state of the rest of the world.
Tim Cain, one of the main creators of Fallout:
Killzig: What ever happened to the catholic church in the FO universe?
Tim Cain: I think it was nuked - the vatican that is.