It's an interesting question. When I did mine in London, the DMP emailed me a few days later with a mail tracking ID so I could see that it was sent to the London visa office. But it didn't show up on e-Cas until 4 months later (1 month after they started processing my file).
I'm not sure if that means it takes a while to process the medical, or if they were just slow in updating e-Cas.