Whatever you put in conf.d folder will have a global effect, Webinoly have support for Custom Nginx rules per site, for example: /var/www/domain.com/custom-nginx.conf, you can read about this feature, here: https://webinoly.com/en/documentation/sites/#nginx
I don't think that Proxy Nginx directives will solve your issue (I may be wrong), because your WordPress sites are not using the Reverse Proxy configuration in Nginx.
What I recommend in this case, is that first of all you should check the PHP logs and of course Webinoly have support for it: sudo log -php, because I think PHP is not responding for some reason. We have two variables for this:
- max_execution_time (/etc/php/7.3/fpm/php.ini)
- request_terminate_timeout (/etc/php/7.3/fpm/pool.d/www.conf)
Both of them are set to a value of 300 seconds (5 minutes), so I think that this time is more than enough to process almost any individual request, for this reason, I think this is not the cause of your issue, but you should consider it.
If you prefer, you can found the PHP logs here: /var/log/php/7.3/fpm.log for more details.
Also, we have an NGINX directive fastcgi_read_timeout (/etc/nginx/conf.d/fastcgi.conf) that we have set to 30 seconds, maybe you should try increasing this value. Don't forget to restart Nginx to changes take effect sudo nginx -t && sudo service nginx reload.
Finally, you should check your server resources, maybe its time to migrate to a bigger server.
Please, tell me if you have any additional question and keep me posted if you solve this issue.