When ATC uses distances, are they using nautical miles or statute miles?
ATC uses nautical miles, this is because it's the ICAO standard documented in Annex 5 - Units of Measurement to be used in Air and Ground Operations, which states that Nautical Miles are to be used for distance and Knots for speed as an alternative non-SI measurement. Kilometers are actually the primary distance measurement because they are the SI standard, however nautical miles are so entrenched they've kept them and there's no termination date. 

The ICAO annex is hard to find because everyone wants you to buy it, however the standards have been copied into this CAAS document word for word if you want some light bedtime reading.