The Arab nations started the war during Ramadan but it was not necessarily in the belief that its religious significance would give them some kind of upper hand, rather it was because it was strategic war timing with the aim of obtaining a fairly quick and decisive victory over Israel.
The fact that the war was launched on Yom Kippur, the holiest day in the Jewish calendar while many of Jewish soldiers were away indicates the timing of the war was more of a tactic to catch Israel's army at its most vulnerable than because it was the Muslim holy month.
Of course, the move did not help the invaders at all but back fired badly against them as Israel quickly recovered, counter-attacked and roundly defeated all three Arab nations and captured even more Arab territory than before.
