Currently I just check the Trade Routes tab every couple minutes to see if the current routes are complete: it would be much more convenient to have either or both of:
1) a sound effect that plays whenever a trade route is completed.
2) a small icon or number over the trading post to communicate the current/max number of routes, and potentially even their status.
Although it's not exactly what you proposed, I think we implemented something that is pretty close and should cover the needs outlined in this post. In one of our previous updates, we added an option to display notifications even if you have auto collect on, and there is a number on the trade route button now (it shows how many available slots you have).
if you turn off auto turn in, you get a notification.