Consent to Travel with Minors

Call the airline. Don't rely on someone here, guessing at what it should be. Obviously a signed letter of consent would be good cover but have no idea if required.

This is what our tour department suggests for tour members traveling with grandchildren, or single parents, etc.:

In addition to his or her passport, any child under age 18 who is not accompanied by both parents must bring a notarized letter, signed by both parents, allowing the child to be taken out of the US and also allowing the adult traveling with the child to seek any necessary medical care. The letter should include the child's legal name and date of birth as shown on his or her passport, plus dates and countries of travel. For details, call the US State Department at 888-407-4747.

See similar discussion at https://community.ricksteves.com/travel-forum/general-europe/legal-forms-needed-for-travel-with-granddaughter.

You should have letters
1. authorizing the father to travel out of the country/consent to medical treatment for the one year old, and a separate letter authorizing travel/consent for the 5 year old, both signed by both parents and notarized and 2. similar set of letters authorizing the mother to travel/consent signed by both parents and 3.a set of letters authorizing you to travel/consent signed and notatrized by both parents.The letters should be specific as to dates and county/countries to be visited.
I found some guidence for the text of the letters at the US State Department website.
The seemingly extra letters fall under the heading of you never know what can happen and it is better to have them and not need them than need them and not have them.

Below is another link from Customs and Border Protection with more information. Although it appears that there is no formal "government" letter, it will tell you what information is important to include in the notarized letter.

https://help.cbp.gov/app/answers/detail/a_id/268/~/children---child-traveling-with-one-parent-or-someone-who-is-not-a-parent-or

Thank you so much! I would kind of rather be better safe than sorry too but the parents kind of seem to be viewing it as a hassle to get a letter notarized. I didn't even think to have them include me for medical decisions, but I will actually have the kids alone for one of the weekends so that totally makes sense.
Thanks!

Not absolutely clear if you are intending to stay any time in the UK or are just changing planes and therefore will not enter the country.

If it is the former, this is what you need to satisfy Home Office immigration rules:

"If the applicant is not applying or travelling with a parent or guardian based in their home country or country of ordinary residence who is responsible for their care; that parent or guardian must confirm that they consent to the arrangements for the child’s travel to, and reception and care in the UK. Where requested, this consent must be given in writing."

So what you need to do from the CBP would also satisfy UK requirements, if you need to.

It will take some time to get the notarized letter (although not sure I would describe it as a hassle). A bigger hassle would be to have your kids denied entry in a foreign land! If both parents are currently in the US, I would recommend getting it done sooner than later.