The reason widget doesn’t work anymore is that Twitter discarded their old API.
If you want Twitter Widget to work again, solution is pretty simple. Just edit file authenticated-twitter-widget/twitteroauth.php
Find line
public $host = "https://api.twitter.com/1/";
Change it to
public $host = "https://api.twitter.com/1.1/";
And save.

Widget should start working again after its cache timeout expires.

https://www.ads-software.com/plugins/authenticated-twitter-widget/