
I have new phone and installed garmin connect app. I logged in my old account, but application can't load badges and list of challenges is empty.
It looks like a connectivity problem. Try an alternative internet connection. WiFi instead of mobile data, or vice versa, or an alternative Internet Service Provider.
Also try disabling the VPN that I see in the status bar. Garmin's webservers are behind a firewall of the CloudFlare service, and for security reasons they may be preventing the access from some addresses hiding their origin. If you cannot access Garmin's website without the VPN at all, due to the governmental censorship, then perhaps it plays some role in it too. Try a different VPN, then too.
I just tried with both Android (v4.55.1) and with iPhone (4.55.2.4). Works in both of them for me, although Android is slower to load the pictures. Either wait if it gets better in a few days, or, if you really urgently need to see the pictures, try clearing the cache of the GCM app, or reinstalling it.