How do these diodes prevent opamp from saturation?
But in this case diodes are preventing such saturation even though the feedback is cut when the FET is off.


When one of the diodes is forward biased, then there is feedback through the diode. 

So in the sample state, the feedback path is through the JFET and the 2nd op-amp.

In the hold state, the feedback path is through one of the diodes. 

Since there's still feedback, the op-amp output will only go to ~0.6 V above the input voltage, rather than to its saturated value (near the supply rail).