Mach number is not dependent on temperature - it's simply based on the ratio of static pressure to total (impact) pressure:
where:
qc is
impact pressure (dynamic pressure) and
p is
static pressure
is the
ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air).
Temperature is needed for true airspeed, but not Mach. The Max ratings are a function of altitude and Mach number, (airspeed does get involved in some of the takeoff calculations, but is not used outside of the Takeoff envelope).
On the CFM, the EEC compares N1 Command to N1 Actual, and adjust fuel flow, etc. to match the actual N1 to the command. In Normal Mode, N1 command is based on thrust lever position (Throttle Resolver Angle or TRA), Altitude (pamb), Mach number, and Total air temp (TAT).