B777 Captain reported having problems with the aircraft data link resulting in their flight number not matching with Foreign ATC displays.

Narrative

After flying from ZZZZ to ZZZZ1 and preparing to return to ZZZZ; We learned that the entire data link system for the airline had been down for 1 hour; and would stay out for an undetermined amount of time. Apparently; this was night 1 of three days that the system was to be purposely down for maintenance of some sort. While the timing worked well for the US/domestic operation (middle of the night); it was right in the middle of the day for the [foreign] operations. The aircraft would not 'Auto Initialize'; which of course; begins the sequence of automatic data links to the aircraft. After a crew discussion; the consensus was to manually input the route.The entire crew was present; and coordinated the correct loading of the route. It was then discussed; that the weights would not data link; as well as any performance information. Ultimately; we were shut down until the data link came back up. Eventually; the system slowly returned; but in a very reduced capacity. The auto init function was utilized; as well as all the performance information: the predicted takeoff data; final weights; and final takeoff data. As a side note; the flying FO (First Officer); did the load at the gate; and the final load after engine start. After a normal takeoff and climb; FO went to crew rest; as he was the Flying Pilot. FO relief pilot took the right seat. Near the fir boundary; at cruise altitude; the Center Controller told us that the ADS-B info did not match our flight number. The language barrier was significant; as this was not a typical ATC conversation; and several attempts were made to confirm what the problem was. After investigating several areas where we may have had a disconnect; I believe that we realized the flight number that would normally have data-linked to the route page was empty. We then consulted the [stored information] to determine the correct format for that input. Initially; I believe we input Aircraft Y; rather than Aircraft X. After the flight info was input; we received no other notification of the flight number mismatch; and assumed that the problem had been corrected. It is my understanding that was the case. In any case; the system reset caused us a great deal of trouble; both here and three days later when conducting the last flight of the trip pairing. Automation is a sensational thing when it works correctly.