No answers from Synology in over 3 months? So many unanswered threads in these forums.
Just to save everyone else the trouble I've finally been able to determine what seems to cause this - custom port mappings.
I've just completed a full format and factory reset which enabled me to start downloads again. I started to apply my old configuration settings back one at a time and checked the downloads after each change.
I have Download Station and Surveillance Station installed. I applied custom port mapping 9092 to Download station. Started a download - still worked. I installed Surveillance Station. Still worked. Applied custom port mapping 9093 to Surveillance Station. Tried to start a download - stuck on waiting, downloads wouldn't start. I removed the custom port mapping from Surveillance Station, stopped and restarted Download Station and my downloads started to work again.
Not exactly sure of the exact cause of all of this, but this seemed to do the trick for me, it may be the same with other application packages when port mappings are applied.
Hope this helps someone else from going through the same frustration I've had the past few days.