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Greetings, Assistance is required about an e90 system whereby, subsequent to the water being heated to a temperature of 80 degrees, occasional fluctuations occur wherein the temperature ascends to 120 degrees and thereafter reverts back to 80 degrees. I have replaced the water pump, thermostat, and water temperature sensor in the N47 diesel engine.

It is possible that there is still air present inside the system.

In the event that the radiator fan fails to engage, it is possible that either the gauge is providing inaccurate readings or there exists a faulty connection, particularly if any components have been recently replaced.
What information does the live data indicate?

The functioning of the fan is seen, and it is peculiar that the temperature fluctuates rapidly and intermittently. Despite the diagnostic readings indicating normalcy, no changes were observed upon adjusting the wiring. The examination of temperature values in the diagnostic analysis suggests that the wire resistance remains within acceptable parameters.

I have not seen this issue with a cold engine; rather, it manifests intermittently only when the ambient temperature exceeds 80 degrees Fahrenheit.

It has been suggested that there may exist pockets of air inside the system.

The thermostat remains in a closed state when the temperature is below 80 degrees.

However, it should be noted that the thermostat had a modification and the whole system underwent a process of ventilation.

Are there any indications of coolant contamination or loss from the reservoir or any other location? (I am considering the possibility of a head gasket issue.)

No, there is no evidence of coolant loss or oil contamination in the vehicle.