If you delete e-mail messages, and you immediately go to check, you will see it as if nothing has been deleted.
You have to give the server time to actually delete all those messages.
That again depends on the home network speed and the amount of messages to be deleted.
It is better to log in to the webmail service and delete the messages from there, then it will go a little faster.
Contact also your ISP.
Check 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
Check Wi-Fi connection, 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
If they are On, turn Off Hotspot & VPN.