You need to enable the error logging in WordPress from the wp-config.php
define( 'WP_DEBUG_LOG', true );
After that, you will probably get some feedback inside:
wp-content/debug.log
Above that, you can try to get the exact feedback you get from 404 page.
Check the console in the inspector, and the response from te webserver.
curl -I http://yourwebsite.com/wp-login.php
Looks like inside your theme you have some redirecting problem. Try to remove the plugins, the theme and to set the 2017-teen new theme to check if this is true.
I hope you will isolate the issue soon.
What I suggest, you create the test environment somewhere and upgrade the website to the latest version of WordPress.
The other things you may investigate: (1) you may have the altered WordPress core somehow, and (2) you may have some .htaccess files other than the WordPress root folder.