OP, I think you need to be really careful here.
If he has made a threat, and the girl is underage, or worse under 16, then a letter of apology won't make any difference to the school's actions. First and foremost, they need to safeguard their other students, and if they believe the threat was credible, then they would exclude him and likely not allow him back on site for exams either. Because if something did happen, and there was already evidence of a threat, that would be disastrous for the school as a whole.
If he writes a letter of apology, and the girl does go to the police, it could be used against him- i.e. an admission of guilt.
I think you need to ask for a meeting with the school on Monday ideally, and ask what sort of action they are planning to take, and try to persuade them the threat was not credible, and that your son is not a risk to female students. You should try your best to find out what the girl's parents are planning to do, although, of course, the school may not tell you.
That said, if the school don't take action, that's often when parents will involve the police, so it is a tricky line to walk.
My experience as a teacher of the police in these situations is that they often will not act without clear evidence, otherwise they just speak to the people involved and say there isn't enough evidence to proceed. If there's a letter of apology/statement that could be read as an admission of guilt, or intent, then that could actually encourage the police to take action.
If the parents do go to the police, and there is an ongoing investigation, it's very unlikely he would be allowed back on the school site until it was resolved.
It is not simple to find an exam centre to take a private candidate at short notice, so I would start looking at local alternative places where he could sit his A-levels. It is likely to be costly.