Is your Smartphone/Tablet on the same Wifi network as the weather station?
Just trying to understand if there is anything out of the ordinary in the setup.
So you are clear the work is done by the smartphone (WSview app)
It uses UDP to query the wifi side of the weather station for settings and if it finds that the published version via the Internet is higher, it (WSView) downloads the new Firmware.
It will then apply this from the app to the Weather station by over writing the old firmware.
So in my simple mind if your smartphone app (WSView) is talking to the weather station and can see it OK but is not seeing a need to update the firmware then this indicates that the WSView app is not able to dial home to the Ecowitt servers for whatever reason. This could be your network (possible firewall settings?) or even your ISP might be to blame.