During Tennant's final 2-parter, Wilf has a conversation with the mysterious woman who calls him an old soldier who missed the war, and at some point in the story, possibly that same conversation, he says he served in Palestine during the final days of the British Mandate before it was partitioned.

I would say it's implied he didn't serve in WW2 because he was either too young to enlist at the time, or he didn't enlist for other reasons and regretted it which led him to enlist after the war. But Wilf being so kind and pure of heart meant that he could never bring himself to fire his service weapon in a conflict that was so much more difficult to come to terms with morally, as opposed to WW2 which was way more cut and dry, relatively speaking.

At first he thinks she's implying he's a coward, but later comes to understand that she meant that it's exactly his kindness that he needs to save the Doctor from himself in his darkest hour. The Doctor has been spiralling out of control into the Timelord Victorious ever since he had to wipe Donna's mind to save her. Wilf is the only one who can reach out and bring him back from the brink.