There seems to be a second listing for father Alec in 1953 in the Voter's list:
1953 Vancouver Center Alec A Plummer, broker and wife Mrs Alex A
Apt 62 1386 Nicola St. (Also living there is a Gordon Wismer, broker, and his wife). Alex Sr (broker), wife and son Alex (marine mate) are at that address in 1940 as well.
In 1965, Alex B and wife Shirley are together on Howe Sound.
As Alex had a brother who died young with the middle name of "Fane", this might be a cousin: On the 1953 Voter's list for Okanagan BC, there is an A. Fane Plummer, with wife Norah.
Googling Alex Bergin Plummer picked up that he was an active T/Lt in the naval reserve during WW2, commanding minesweeper HMCS St. Joseph from 4 May 1944 to8 Nov 1945. As this ship was built in Nanaimo, I'm just guessing that he was just working the west coast of Canada, and not overseas (but I am just guessing!).
None of this particularly helps, but it does help form a time line!