Have you read:

J.R.R. TOLKIEN: A BIOGRAPHY by Humphrey Carpenter
THE TOLKIEN BIOGRAPHY by Michael White
THE HISTORY OF MIDDLE-EARTH SERIES, edited by Christopher Tolkien, with close focus on Volumes VI-IX, which cover the writing in exhaustive detail of THE LORD OF THE RINGS.

Since this is what I'm going by.

But for what it's worth, what Tolkien did in WWII:

* He did not engage in any kind of war work for the government. He was asked due to his knowledge of German, said no, and the government agreed his skills were not essential and let him get on with his life.

* Acted as an ARP Warden in Oxford, which involved relatively little work as Oxford was never bombed in WWII.

* Kept some chickens in his back garden to aid in rationing.

* Engaged in minor work at the university, giving basic English tutoring lessons to military cadets. This was a drop in the ocean compared to his normal tutoring schedule pre and post-war, which was extremely busy.

* (Right at the very end of the war) bitching at C.S. Lewis for ripping off Middle-earth in some kids' books he was planning to write about a magical wardrobe or something.

In fact, the most stressful period of the war for Tolkien was probably 1943-44, when his son Christopher was stationed in South Africa being trained as a pilot (Tolkien considered aerial warfare to be amoral and obscenely dangerous). And that was the most productive period of writing on LotR, when he wrote almost all of Book IV of LotR (Frodo and Sam meeting Gollum and sneaking into Mordor) in just a few months of frantic activity.