So if she wasn't entitled to diplomatic immunity then she has simply left the country which means the police can apply for her to be Extradited. This is what happens, it is the law as simple as. There is no need to go on a national media campaign, no need to go to America and no need to be on television every day which certainly cant be good for them.
That only came out as they started digging. It sounds to me and this is only an assumption of course, that someone thought, " Simple working class family, we'll tell them she was covered by diplomatic immunity and they'll accept it and go away." Then when this news came out they put it to Trump who being Trump and viewing it as an attack on America starts the "She won't be extradited " rhetoric. I say good on them and who cares if they get the media behind them? Their son was killed by a woman driving on the wrong side of the road and hit head on. She said she would cooperate with the police and was doing so until she realised she might be in bigger shit than she thought and fled.
They went to America to put pressure on. Trump has a God like complex, he has already pardoned people he thought shouldn't be imprisoned so I wouldn't set much stall out with staying at home and waiting for the law to take it's course.
Everyone deals with grief differently. I found by keeping busy it helped. The whole world now knows about their son's story and therefore massive pressure is heaped on the Yanks to do the right thing and get the woman back to the UK to face up for what her negligence caused, the death of another Hunan being. I'm sure they'd rather have their son back alive and go back to being unknown Mr and Mrs suburbia but unfortunately for them that isn't going to happen.