In the Star Trek universe, it's not completely clear to me who the belligerent factions of World War III were. Who were they and what is known about them?
I'm going to shamelessly steal from Memory Alpha:
Among the parties involved was the Eastern Coalition (also referred to as "the ECON"), whose direct attacks included those against the United States of America. (Star Trek: First Contact) In 2026, at the start of the war, Colonel Phillip Green led a faction of eco-terrorists that was responsible for the loss of 37 million lives. (TOS: "Bread and Circuses", "The Savage Curtain"; ENT: "In a Mirror, Darkly, Part II")
NON-canon material (books) elaborates. From Memory Beta:
One side was composed of the United States and the European Union; the other side was an alliance known as the Eastern Coalition.
The detonation of nuclear weapons over cities such as London, New York City, Tel Aviv, Jerusalem, Mecca, Riyadh, Karachi, and New Delhi killed nearly half a billion people instantly.
During fighting in Paris during the war, the Eiffel Tower was destroyed. (DS9 novel: Trapped in Time)
So, it was US, EU (obviously new to the material since the age of Roddenberry) - including at least UK and France, India and Israel all involved on one side.
Please note that there just aren't all that many details known about WWIII in Star Trek:
... very few details have been presented in Star Trek on World War III. In 1996, this event received a Trekker's Choice Award for the 'Oft Heard but Never Seen' Award, as being "the favorite historical moment only alluded to in Star Trek."