Gulfstream G650 Captain reported problems with the CPDLC equipment while in contact with Shanwick Control.

Narrative

We had just passed 52 degree North and 30 degrees west. Our CPDLC equipment was to automatically switch over to Shanwick Control. It did not. In fact the automatic handoff locked up the box. We received a SELCAL on HF to call Shanwick radio. The radio operator passed a message from Shanwick Control to reset our CPDLC Equipment. When we attempted to do that; the result was the CPDLC taking down and locking up two of our three FMS's. Fortunately the number three FMS never failed. We continued navigating with that FMS as the navigation source. Within a very short time the number one and number two FMS's rebooted themselves. The CPDLC unit which is tied into all three FMS's regained some functionality; but did not come back to normal operations until after we landed and the aircraft was totally shutdown and restarted. There is no way to turn the CPDLC unit off in flight; the FMS's were also given a byte drop to clear out their internal memory after downloading same.Honeywell and Gulfstream are investigating the problem; with no answer as of yet. But they admit this has happened before.