Have you considered trying to renew the kids' passports? There's an online premium service, assuming your kids have (or are entitled to) British passports. But you would need to apply today to have any chance of getting the passport for Tuesday. EDIT: The online premium service appears to only be possible for adults.

If the kids have, or are entitled to, a passport other than a UK one, then check with that embassy or consulate if they can issue an emergency travel document before Tuesday.

For future reference, you should always renew passports before they get into their last 6 months (if you are expecting to travel and use them within that period of the final 6 months' validity), because that's when airlines start getting iffy and anxious about accepting them. Even if there *technically* is no need, you have seen first-hand that it actually saves you time, money and worry. And in most cases, the additional time is added to the new passport, so it's not that you needlessly pay for an "overlapping" period across subsequent passports.