Their families and friends helped them, Brandy as families and friends still do when people are depressed. The difference is that there are more medications available now as well.
Clinical depression (which I think you are talking about) is misery which fails to lift even when circumstances improve. Most of the depression of the war years was unhappiness with a definite cause. Fathers, sons, husbands, lovers, brothers, friends were in danger. Houses were being destroyed and treasured possessions being lost. Sleep was broken by night bombers and worry about the very real possibility that there could be a telegram in a yellow envelope. Food supplies were scarce and erratic. Children were sent away from unsafe areas to places far from their families. There was plenty to be unhappy about.
One thing which kept people going was the knowledge that others understood because they were under the same strain, and there was always someone worse off. That was not a cure, but it was a support.
"Counselling" was talking things over with a sympathetic friend over a cup of tea. Some friends were better at this than others.
"Medication" was barbiturates which dulled down the feelings of misery and enabled people to sleep. Severe cases were hospitalised until/if they recovered.
Don't underestimate Vera Lynn's schmaltzy songs. They were a real comfort to many, articulating the feelings which were overwhelming both those fighting abroad and those they left behind at home.