I'll let someone else address your question, but I did want to point out some potential pitfalls.
Your kids must enter the US on their US passports. However, they can't then enter the UK on those American passports unless they have a visa for the UK.
Be sure to bring both.
Personally, I wouldn't bother with a letter for my own kids. My wife has made tens of trips with the kids and without me and it has never even been a minor thought in anyone's mind. Why should it be different for a man? Not every man is a child molestor or abducting their own children and the people who think this should be challenged. Not having a go at you, OP, it's a valid question
That said, I must have travelled internationally with my kids a few times and never had a problem. My 15 just went to France by herself and wasn't asked anything.
Not having a go at you, OP, it's a valid question.