I would be willing to bet that at some point right before the outbreak started, there were at least minimally measurable amounts of NO3 and/or PO4. What can very easily happen is the cyano (or any nuissance algae) consumes those available nutrients in the water, leading to test results showing 0s.
This is what's happening in the 50g reef we set up at work (a very young tank). There were NO3 levels in the 10 - 20 ppm range a few weeks ago. About two weeks ago, a red cyano outbreak started. Now, there's plenty of red cyano and 0 ppm NO3. I work to remove the cyano as much as it's reasonable, to prevent any nutrients from being recycled within the tank as the cyano begins to run its course. It takes time to battle it like this, but it's been effective for me in the past.