Hello and thanks for your question.
Your son cannot take his daughter out of the UK without a court order or pernission of the mother. He does not have a residence order only a contact order and from what you say the contact order does not cover this scenario. Your son could make an emergency application to court today or 16th June - but it is pretty unlikely that the court would agree that this situation counts as an emergency. The court forms needed are C100, and C2. The C100 sets out all the details and the C2 is to ask for an emergency hearing at very short notice to the mother (instead of the usual 14 days notice, ask for 24 hours notice to her).
Here are the forms:
To have any chance at all, I would strongly recommend that he gets to see a solicitor very quickly indeed to prepare the appplication to court, and to write a strongly-worded letter to the mother.
Here's where to find a specialist family law solicitor:
You are quite right in your approach - that the application has to be worded about the effect on the child, because in court cases concerning children, the court decides on what is best for the child.
I know this is not the answer you were hoping for but I hope it helps anyway.
I wish you the best of luck.
Thanks and best wishes...