The Chicago Convention does not apply to state aircraft. Therefore the standards relating to the format of call signs contained in Annex 10 to the Convention do not apply to US military flights. They are therefore entitled to give their aircraft whatever call sign they want. It appears from the Times article that the flight plans were being marked up as state flights so there does not appear to have been any attempt to pass these flights off as Jetsgo flights. That would have been especially hard in the case of the flights involving C-130s and MC-130Ps.
More generally I am not aware of anything in UK legislation which specifically addresses the use of inappropriate call signs.