"There was an error connecting to the Apple ID server"

Try first to optimize the range of the mobile network, see this Apple article:

If you see No Service or Searching on your iPhone or iPad

If you see No Service or Searching on your iPhone or iPad - Apple Support

Try to optimize the Wi-Fi network, see the following Apple articles:

Connect to Wi-Fi on your iPhone, iPad, or iPod touch

Connect to Wi-Fi on your iPhone, iPad, or iPod touch - Apple Support

If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network

If your iPhone, iPad, or iPod touch won't connect to a Wi-Fi network - Apple Support

Recommended settings for Wi-Fi routers and access points

Recommended settings for Wi-Fi routers and access points - Apple Support

If you need help with your Wi‑Fi password

If you need help with your Wi‑Fi password - Apple Support

Then try again to sign in.