What happens to EGT after a bird strike?
No two bird strikes are the same when it comes to engine parameters, we can look at this incident.

Bird strike in both engines, yet only one showed above-normal EGT.

In a jet engine, the bypass air assists in cooling, damage will disrupt this air flow.

The core might still be running, or flammables (oil, fuel, etc.) might leak and catch fire.

In an engine fire, EGT-rise is one of the confirming factors.



However, according to the NTSB report for Sully's plane:


  No in-flight or postaccident fire occurred.


So if it is just an engine failure, the engine will spool down.

Even if the damage prevented the windmilling, EGT should still drop. If the EGT sensors were damaged, an indication of EGT reading failure is to be expected.

Also the engine vibration indicator should show very high vibration after the bird strike.



(Source)