Know the url of File being downloaded from App on Android phone
One way would be to install a network packet sniffer app; there are some that do not require root, by acting like a VPN app.  You seem to be technically savvy since you mention IDM and uget so what you would also need besides the packet sniffer app is software to analyze the resulting capture.  Generally, once you capture the data you would transfer it to a PC then run a network capture protocol analyzer; such as Wireshark (available for Windows, OS X, and Unix) to analyze the traffic and you should be able to find the URL of the file.