Assuming that you've submitted a Delay Message in order to prevent your flightplan from being deleted you have to do nothing at all.
In your flightplan you only state your ETD (Expected Time of Departure) and ETE (Estimated Time Enroute). Your ETA (Estimated Time of Arrival) will be calculated and relayed to all stations in need of the ETA by AIS (Aeronautical Information Service).
To pick up your example:
Filed ETD: 1500
Filed ETE: 0200
AIS will relay ETA 1500+0200=1700 to your aerodrome of destination.
Let's say you've departed at 1554. Your aerodrome of departure will pass a departure message to AIS (so they know you are on your way) together with your ATD (actual time of departure) which would be 1554 in the example.
AIS will then calculate ATD+ETE => 1554+0200=1754 and relay the corrected ETA to your aerodrome of destination where the data in your flightstrip gets corrected.