Coolant temp gauge bouncing indicates intermittent CAN-bus connection from DDE (engine control unit) to instrument cluster (gauge goes to max if there is no CAN information when instrument cluster has ignition pos.2 power). Auto gearbog is also in the same CAN-bus. Like suggested check and clean the connector on auto gearbox (gearbox ECU is inside the gearbox). Just in case also check the E-box near firewall where DDE is located for possible water damage.

When doing basic diagnostics it's difficult to cause any more harm. So check the codes if you have any means.

How high did the water reach and for how long was the vehicle submerged?

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E39 530dA -02 M-Sport Messing metallic
E53 X5 3.0dA -06 Sport Stratus grey
E70 X5 40d -12 M-Sport Space grey