I discovered that there is a nvm plug-in shipping with oh-my-zsh (that's different from lukechilds plugin). After short inspection, I think it adds the necessary modifications to .zshrc when loading, so simply adding nvm to the plugins list in .zshrc should work as well (and it does for me).
I did not find any more details on that default nvm plugin via google so I don't know whether this is the "go-to" solution.
If you already have nvm installed and you're switching from bash to oh-my-zsh you can simply open up your .zshrc file and add the nvm plugin that is included with oh-my-zsh:
Open your zsh config file.zshrc in nano with this command: nano ~/.zshrc
Scroll down to where it shows plugins=(git) and add nvm inside the parentheses to make it show as plugins=(git nvm) (separate plugins with spaces)
Press control + O (on macOS), then enter, to save, then press control + X to exit
Then open a new terminal window/tab and enter nvm ls to confirm it works. Note that you must open a new window/tab for your shell to use the newly updated .zshrc config (or enter source ~/.zshrc, etc.)