You are rewriting the root to a fixed API slug, I’m not sure that’s what you want, the rewrite isn’t very flexible, try:
rewrite /(.*) /$1 break;
I think this is what you had before for your gateway, what you want to achieve is that everything gets proxied to Tyk for processing, you need the listen path to reach Tyk.
You may be correct in assuming that Tyk doesn’t have an API to listen to, in which case it will always 404.
If there’s no nodes listed in the dashboard, I have a feeling Tyk isn’t bootstrapping and registering, so I’d suggest tailing the log files of the gateway (/var/log or /var/log/upstart).
You’ll need to solve that before your NGinX rewrite.