The compression ratio is just a numeric value that represents the ratio of the whole cylinder, including the combustion chamber, when the piston is at the bottom of its stroke, when compared to the volume of the combustion chamber when the piston is at the top. Any volume below the head gasket, like piston dish, is included as part of combustion chamber volume & so is the volume that is encompased by the head gasket.

The cam timing has no bearing on this arbitary compression ratio number, but it doesn't take Einstein to realise that the cam timing has a very serious effect on actual compression 'pressure'. A wild cammed engine may have 12 to 1 compression ratio, but if the inlet valve is still open with the piston part way up the bore, then at low speeds it is pushing the inlet charge back out through the carbs. If the valve actually closes at, well,... say,.. half way up the bore, then the amount of gas actually being compressed is only half as much as the mathematical volume of the cylinder. Under this imaginary scenario, the actual compression ratio, as seen by the combustion chamber, is quite a bit less than the theoretical value.
The same goes for unmodified engines at idle. The valve may well close at a more sensible time, but the volume of gas actually in the cylinder is quite small. It would have been at a lot less than atmospheric pressure when the valve closed, so the piston may well travel a fair way up the bore before cylinder pressure rises to be the same as the atmosphere outside the cylinder. It is only at this point that compressing of the fuel charge actually takes place.

The engine actually sees the full compression ratio as 'maximum cylinder pressure' when it is developing maximum torque, so in a way, all I.C. engines are of the variable compression ratio type.