Hi Squid,
First, you're not being unreasonable in your frustration. Second, I think you should focus more on the stress and frustration on you and your family with this new schedule. I've done this airport change before and it's not fun. I took the National Express bus and needed every minute of the 3hours I had to make it to my flight. I was one of the last people getting on my plane at Heathrow (to Seattle).
There's only so much that BA will do for you in this case. I have had great luck with everyone that works at BA, but I've had the best experience getting service through their Twitter agents. I think there's something about getting things in writing that makes it so smooth.
Here are my suggestions:
If these are award tickets read number 1 or if they're purchased tickets read number 2.
1. Since these tickets are mileage awards then I would consider one of these alternatives:
a. Keep the flights as booked, but plan that if you miss your flight LHR-DEN that you may be rerouted to a later flight or more likely have to stay in London and fly out the next day. I would imagine this would be stressful for you and your family.
b. See if BA might change your flights: Fly to London a day early. This option has the most hotel and transportation costs, but would be more of a sure thing and possibly least stressful.
c. See if BA might change your flight to depart from another Italian city: You could take the train the night before to Milan and fly out from there instead. There are flights from Milan to Heathrow that would work for your Denver flight.
2. Did you buy these tickets?
If you purchased these tickets I would ask for a refund and book your flights on Lufthansa instead. Lufthansa has a non-stop flight to Frankfurt from Denver that will work well for you.
Let us know what you think. And don't forget, BA's twitter customer service people are awesome.
Good luck,
Nordheim