A practical answer:
If you added the $^{60}$Co the way you dont shield the Cs and the rate from Co itself is not huge (e.g. it is like 100 cps):
you will not be able to detect else than 100 cps in the 662 keV peak (the dead-time will not change in an important way).
2- If you added Co with a high countrate (several thousands or more), you will experience stronger dead-time effect and you will see less of 662 keV. Supposing you have an analog acquisition system and HPGe detector, 100 cps does not make an important increase in the dead-time.
3- If your system is especially slow or bad tuned (e.g. you have a high dead-time at low countrate), the results are not too much predictable.
A theoretical answer:
Every time you add more countrate to your system, you basically remove the time for the readout and conversion - you increase the dead-time. It is possible to imagine an experiment, where you would like to study this effect. For a case of conversion time 20$\mu$s and rate 100 cps, you would need to have more than $10^6$ pulses in the peak to be in the range (with statistical errors) that would allow to look for the dead-time effect. But real problems would only start here.