This is something that is an easy fix. Nginx is a great web server not only for your primary site but for a proxy as well since it is made to serve a lot of requests very fast.
So it is as simple as this
location / {
proxy_pass https://staging;
proxy_read_timeout 500;
proxy_next_upstream error;
break;
}
The key lines are proxy_read_timeout and proxy_next_upstream.