About luck and bad luck, how to say in English

There is no locution in English which is the equivalent of "war luck." You would need, therefore, to say: "That was the time when luck in war turned to his advantage." ("Turned to his side" would also not be a common locution in English. "His side" would be taken to mean him and all those who adhered to him, or to his policy.)

People talk about the "tide of fortune" turning against people, e.g. against Hitler from around 1942.