Hi, goingtoitaly. I haven't gone from England to Italy, but I've done London-south of France (or Scotland) by train, and around Italy by train.
My verdict on the journey to Avignon (2 hours to Lille, then 3h 30) was that train is the only civilised way to travel with small children - you get to St Pancras, and then go through simple security and ticket check 30 mins before departure. Much better than an airport, and quieter and less claustrophobic than, say, Gatwick's jam-packed check-in area. Then once on the train, more space to move about, can jump about between cars, and there's fresh air and no plane noise, and you can bring your own picnic lunch. And no worries getting from airport into the city, because you're already in the city! Better scenery than the plane, too.
There are trains where you can change in Lille/Brussels/Paris and end up in Northern Italy, but you'd probably want a sleeper. Even if you don't get the best night's sleep!
The Eurotunnel is only 20 minutes, and if you can distract yourself for that long, it would save you having to look into ferries which would make everything a lot more difficult.
Italian trains are cheap, although you may need to pay a 3-6 Euro supplement on some for reasons I never understand. The Milan-Venice line is great, stopping at Verona. From Verona you can get a bus to Lake Garda which is a lovely place for a family holiday. The trains tend to have 3 steep steps into them, but people will help you.