US citizen travelling to UK with expired US passport for a minor and valid UK passport

Legally all US Citizens must depart the US using a valid (non-expired) US Passport. This includes people with dual citizenship.

Thus your daughter, a US Citizen, can not legally travel to the UK using only her UK passport - she must do so using her US passport.

As the US does not have any (physical) outbound immigration it is possible that she will be able to depart using only her UK passport - but as I said this is not legally allowed.

I don't know why she would not be able to check-in online. On many airlines, including United, it is not uncommon for foreigners (as she is if she's using a UK passport) not to be able to complete the on-line check-in process for an international flight until the airline has physically seen your passport. When that happens with United you will be able to complete check-in, but instead of receiving a boarding pass you will recieve a document that says "This is not a boarding pass" and has instructions to see an agent at the airport - but that doesn't sound like what you're hitting here. A UK citizen would obviously not need a visa to travel to London, but perhaps their system is smart enough to realize that she is a US citizen and thus has to use a (non-expired) US passport.

So really she has 2 options - either go to the airport tomorrow and attempt to travel using only her UK passport (which is illegal, but may work), or delay the trip and obtain a new US passport.

Like Doc said, legally, all U.S. citizens must enter and leave the U.S. with a valid U.S. passport. However, the U.S. has no exit checks, so one does not have to actually do anything to satisfy "leaving the U.S. with a U.S. passport". Theoretically, the CBP could perform random checks on outbound passengers, but I have never seen that happen. I have also never heard of any consequences for someone violating the rule when leaving.

An airline can only, and should only, care about how you enter the destination country. They care about how you enter the destination country because they are liable to transport you back if you do not have permission to enter. But there is no point for the airline to check whether you can leave the originating country, even in countries that have exit checks, because if you can't leave, then you don't fly, and it doesn't hurt (might even benefit) the airline.

Even people who have no status (or had never had status) in the U.S. are free to leave at any time, without encountering any problems. So I doubt your daughter will have any problems leaving the U.S.

Theoretically, the embassy in London could get pissed at her for leaving the U.S. when her U.S. passport is expired, but it's not likely to come up, because they would first have to figure out when she left the U.S., etc., which is only found on her other passport. Plus, for all they know, she could have never realized that her passport was expired, since it is never checked on exit; or she could have never realized such a rule existed in the U.S. (I'll bet most U.S. dual nationals don't know).