Your suggestion is right on. Changing diff or tyre size leads to incorrect speedo meter and odometer readings.
You don't have to convert KMs into miles, as they will be incorrect as well by same ratio as KMs.
As @capsat suggested, use smartphone and Sygic GPS software which displays current speed as well to see the difference in speedo meter reading. For example drive the vehicle and note speed from Sygic GPS, go at 60KM/h speed now check speedo meter, it will be reporting probably about 70KM/h or so. Note it down. Do couple of more of these same tests at different speeds and note down differences.
Now you will have a rough idea of actual speed than reported by vehicle, and you can also calculate ratio based on these recorded values which you can multiply with current speed of vehicle or odometer value to get actual speed and odo reading without help of GPS.
Hope it helps.