How can I configure an Ubuntu server installation to provide internet access to private network, such that machines on the private network can access the internet through the Ubuntu server? The machine has two NICs of course.
Unless you are prepared to deal with a lot of automated attacks on your server, I would highly discourage this. Your router will provide a level of protection between you and the internet.
What exactly are you trying to accomplish? There might be a better way to do what you want. I have my ubuntu box acting as a DHCP server with dockerized Pi-Hole installed. It blocks a good majority of ads and allows me to fine-tune DHCP settings that the router won't let me.
If you really, really need to have the server facing externally, I would recommend that you harden the server significantly by denying all access to the internal LAN unless the request originated from within the LAN.