How could I configure nginx so that when a download is started and when
it completes, it makes a HTTP GET request to another web server?
This is not possible.
You need to write a custom Nginx module.
I think you should be able to use the ngx_http_echo_module to achieve
this. Just use an echo_subrequest directive to send a request to the
URL that you want. This request will only be made once the upload has
completed. Any other module that calls a subrequest after the files
have been uploaded should work too.
One thing : if you don't know the server that you'll connect to at the
time the config file is read, you'll need to set up a resolver for DNS.