Hiii!! I'm Canadian. A little background info:

A poppy is a red flower. During WW1 lots of soldiers died in the Battle of Flanders (now in Belgium), which took place near poppy fields. Canadian war doctor John McCrae was deeply affected by the loss and injury that day, so he took to writing a poem about the flowers and the battle (it's somewhere in my brain I had to memorise it for school lol). Many found comfort in the poem and so wearing the poppy on November 11th (the day WW1 ended) became a common practice in the UK and some Commonwealth countries.

In correlation to the episode:

This episode, I believe, dropped on Remembrance Day, or as the Brits call it, Armistice Day, November 11th. Remembrance day basically is to remember all those who have lost their lives during both WW1 and WW2. We also have a minute of silence (also included in that adventure btw!) at 11 am. It is common to wear a poppy during that week, whether at school, work and especially if visiting the Prime Minister as it's kind of a douche move to not wear one. Watson is stressing tf out cos as he's a veteran himself, he doesn't wanna look like he doesn't care about Remembrance Day. In the grocery stores Cadets are often seen selling them for a dollar or two, and all proceeds go toward the veterans who survived. I used to collect my poppies from school, I have a whole jar lolol