At an in-principle level, no, it doesn't change the laws of physics, but it could change how we write the equations we use to describe them.
The laws of physics depend on the dimension Time. Any given clock is simply an instrument to measure time. The standard scientific unit, which we measure in, for time is the second.
The second is currently defined based on the vibration of certain atoms, but was originally defined as a fraction of an minute which was in turn defined as a fraction of an hour and so on - ultimately the second was originally based on the rotation of the earth. This could be considered one "clock" in the way you mean, if i understand you correctly.
Changing the definition to be based on atomic vibrations allowed a more accurate definition of the size of the unit, but didn't change any fundamental physics, which ultimately relate to the dimension, time. Choosing to define it based on the vibrations of a different atom wouldn't have changed anything except the ease/accuracy/reliability with which we could measure the unit.
If you mean choosing a different unit size, other than what we currently know as a second, that would either have resulted in different choices of other unit sizes, or different looking fundamental equations - but they would only have differed by the constants needed to make the units work.
Take Newton's Second Law. The fundamental law is that force is equal to the rate of change of momentum. Given constant mass, that results in the familiar
F = ma
This is given use of kg for mass and meters per second per second for acceleration. However, if we wanted to measure acceleration in miles per hour per hour we could. The equation would then be
F = Kma
Where K is a constant (approximately 8052.985) to make the units work. The fundamental physics is the same.
In this case, we pretty much define the unit of force (Newtons) so that this holds (i.e. so that the constant K = 1 and can be left out altogether)
In practice, it would probably have resulted in changes in some other units to make things nicer, rather than just shoving in a bunch of constants.