It didn't used to be so, but lately I have heard only too many tales of dual citizens being detained and "questioned" for up to an hour or more if they try to enter the US on a non-US passport. Assuming that the children are US citizens (there are some requirements for you to be able to pass on your US nationality), they probably won't be denied entry, but you could be in for a long delay at Immigration on arrival.
To get them US passports, you'll need to report their births to the US Consulate in London (which may require you and the kids making a personal appearance in London) in order to apply for their US SSN and passports.
In your case, because the kids have non-US birthplaces, you may be able to get by on their UK passports if it isn't obvious that you are travelling with them and that you are their parent (i.e. that they are travelling with their British parent only).
I don't know about the monetary penalty for not registering your kids for a SSN. That's a new one on me. The one caveat is that, for a US citizen to apply for their SSN from overseas can be a real hassle as an adult.