Is HTTPS behind reverse proxy needed?
It all comes down to how secure or paranoid you'd like your implementation to be. It may also depend on the type of data you're playing with. For instance: I'd definitely do this for credit card numbers or other sensitive information.
As the comments have already stated, you would typically terminate SSL connections at the front facing webserver, assuming the API backend is also inside your LAN, which you trust and control. If you want to go that extra mile, you could also set up SSL on the API backend. Details of how to do that depend on the software you're using on your backend.
If you do decide to implement SSL on the API backend, the setup would be similar to what you did to setup Nginx with SSL on the frontend, with the main difference being you don't need to use a public certificate on the backend. It can be self-signed, since no one else besides your web server will be talking to it. Then it's just a matter of fixing all the URIs in your code to use HTTPS.