How long should one wait for a reference before writing to the course convenor?
I'm going to answer this question myself for the benefit of people who may stumble upon this in the future. 

In the end I didn't email the convenor, but I phoned the Prof in question. I initially thought maybe I needed to give it longer than three weeks and I was being impatient, but it turned out that they totally understood and said they had been very busy but of course they would do it. They ended up submitting the reference the same day. To those people who may be in a similar situation: don't be too shy about phoning or visiting your professors for this reason. They understand the necessities of getting it done and done in good time, but understandably get waylaid by priorities.