How does Mach number of Jet engine vary with respect to Compression Ratio?
The performance and exhaust Mach numbers depend on very many variables, to name a few:


Inlet shape.
Compression ratio.
Combustion inlet velocity of the airflow.
Throttle setting
Turbine entry temperature.
Exhaust shape.


A multi-dimensional matrix which to fully describe takes a book like this one. My edition has 491 pages, and the scope of the answer would be very broad indeed.

The exhaust Mach number however, is mainly determined by the shape of the exhaust. If the exhaust is convergent only, maximum Mach number of the exhaust stream is M = 1. In order to reach supersonic exhaust velocities, the shape must be converging then diverging. If converging only then depending on the operating state the exhaust can be choked, at a higher pressure than the environment. The higher pressure will still contribute to thrust.



Figure above shows an example for a supersonic exhaust for a low bypass engine.