If he was in the military I presume military police has some jurisdiction.

I would also presume there's some mutual law where if a foreign military member is on another countries soil (if you're in the military you can't just go on holiday to a foreign country without extra paperwork) that allows counties to retain their military courts jurisdiction even on foreign soil.

That doesn't mean he gets off - just that the trial would be under US MP law. To be honest, the consequences of which would be higher for him than under UK law.

I believe they forgot to mention she, AFAIK, has a doctorate.

There seems to be missing explanations here. The CPS are the ones to decide where to pursue a case, not the police.

But then there's a major issue of the judiciary and police not really doing their jobs properly, so it doesn't surprise me that this happened.