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Hosting cannot reach Telegram
Some hosts block api.telegram.org, for example many servers in Russia. The plugin then cannot send anything: the settings page shows a warning, and the log shows that your server could not reach api.telegram.org.
Ask your host
Ask your host to allow outgoing HTTPS connections to api.telegram.org. This is the only address the plugin connects to.
The warning on the settings page goes away by itself once Telegram answers.
Send through your own address
Notify for Telegram Pro can send through your own Telegram API address: a proxy server you run outside the blocked network. See What Pro adds.