'N/sec/C' is using the base units for thermal conductivity for the measurement system, and is equivalent to the 'composite' unit you are referring to as the 'actual one':

1 Watt = 1 Joule/second = 1 (Newton*meter)/second

1 Kelvin = 1 Celsius (in magnitude)

1 Watt/(meter*Kelvin) = 1 (Newton*meter)/(second * meter * Kelvin)

= 1 Newton/(second * Kelvin) = 1 Newton/(second * Celsius)

so

1 W/mK = 1 N/sec/C

For the Heat Capacity, it is very important to note that many numerical tools use _volumetric_ heat capacity, not specific (mass-based) heat capacity. E.g., you should be comparing to J/m^3*C, not J/g*C. you convert from one do the other by multiplying by the material density.

Converting:

1 Joule / (meter^3 * Celsius) = 1 (Newton*meter)/(meter^3 * Celsius)

= 1 Newton/(meter^2 * Celsius) = 1 Newton/((1000 mm)^2 * Celcius)

= 1e-6 Newton / (mm^2 * C)

1,000,000 Joule / m3C = 1 N/mm2C