ATC and/or the military could have rung up BA Ops and asked them to ACARS the aircraft in question
I've seen that done in a slightly different way - ATC asked a "same company" aircraft on freq (i.e. us) to contact base to initiate the above string of comms.
I guess doing it that way saved ATC ringing directory enquiries and also the cost of an international call....
Edit to add: Genuine query - as anyone here actually used CPDLC with Budapest?