All incidents should be available through the API the moment they are reported in Nintex Process Manager.
I reported this to support who had confirmation from Nintex Process Manager development team that there is a scheduling process that runs every 12 hours in the backend to further update and show the data in the reporting API from your site.
This seems like a bad architectural decision as the emails come out near instantly, so why couldn't it also be available through the API at the same time?