The UK immigration officials were, and indeed would like to continue to be, very reasonable people for those with documentation difficulties. Unfortunately they were continually being taken for a ride by nationals from certain countries (the USA was definitely not one of these) with a variety of tricks for getting into the UK when not entitled. Hence it has been passed down the line to the airlines etc that if they carry anyone to the UK without the correct documentation they can be fined £2,000 for each such passenger. As a result the airlines just will not board anyone without the right documentation at check-in. Even this is not foolproof - there are constant cases of those who board with valid documents but then tear them up and flush them down the aircraft toilet, and then on arrival claim to be from somewhere else.

Even the "emergency service" for things like travelling to family funerals at short notice was abused in this way, so I am afraid you have to blame those who have gone before you for all this.

All this is from the arriving in the UK view - others here are commenting on requirements about departing from the US, particularly with children.

As you seem to have been able to change your travel plans to bring the date of travel forward, I am afraid you will have to change them back again if the passports cannot be received in time.