This seems to imply that there isn't any high-resolution precipitation data available that could provide these "minute-by-minute" forecasts, but that isn't true. The National Weather Service provides several radar products that give data with resolutions in the range of 500 m using their NEXRAD technology[0]. This allows for some pretty good estimates of when precipitation will start and end over the next hour or so. This kind of forecast product is called a precipitation nowcast. Other nations have similar systems.
If you use the NOAA desktop tool[1] to view the data from NEXRAD stations, you can compare to services like DarkSky and see that they are very likely using it without much editing.
The simplest nowcasts use optical flow techniques rather than meteorological modelling. On short time scales (less than an hour), these methods can give passable results. I built a tool[2] that pulls this NWS data from their Web server and gives you a nowcast.
If you use the NOAA desktop tool[1] to view the data from NEXRAD stations, you can compare to services like DarkSky and see that they are very likely using it without much editing.
The simplest nowcasts use optical flow techniques rather than meteorological modelling. On short time scales (less than an hour), these methods can give passable results. I built a tool[2] that pulls this NWS data from their Web server and gives you a nowcast.
[0]: https://en.wikipedia.org/wiki/Nexrad#Super_resolution
[1]: https://www.ncdc.noaa.gov/wct/
[2]: https://github.com/bmgxyz/threecast