Since this is a European train forum I might suggest to do the whole journey by train from the UK!

There are beautiful Alpine railways and it would be a shame to miss them. For example the Bernina Railway fits perfectly on a UK-Italy itinerary. A relaxing way to cross the mountains after high-speed trains through the French countryside. See: 

https://www.seat61.com/trains-and-routes/bernina-express.htm

https://www.seat61.com/trains-and-routes/bernina-express.htm#Bernina_Express_to_go_from_London,_Paris_or_Zurich_to_Italy (incredible website btw)

Also possible to take a direct Paris - Milan TGV but the views aren’t as gorgeous in my opinion. The route is closed this year due to a landslide but should have reopened in 2025.

I’d make my way slowly south, especially as the weather will be better up north at the start of the month. The Tuscany region is really cool and there are plenty of day trip opportunities.

At some point you’ll have to cross the Messina Strait, do it using the last European train-ferry!

https://www.seat61.com/trains-and-routes/trains-to-sicily.htm

For the way back you could take the convenient night train to Milan (from either Siracusa or Palermo). Of course you could fly back from Sicily if you prefer.

These are some guidelines which shouldn’t change much until 2025.

Fly into Rome this is a major airport and a good starting point for exploring central/western Italy. Spend a few days seeing sites like the Colosseum, Trevi Fountain and also you can take the train to Naples it is the centre for pizza, mozzarella and coastal towns. and also consider spending time in Sorrento, a charming town with beaches and hiking trails suitable for kids. 

With 2 weeks, you could do a great trip flying one way (maybe to Palermo?) then working your way back up to the UK.

The children get free rail passes, so you’d only need to pay for the adults. Plus any compulsory reservations along the way.

I did this last year. I flew to Palermo and spent a few days on Sicily. You can then take day trains (7.5h) or a sleeper to Naples, have a few days there and in Rome. Then a couple of hours to Florence, and use it as a base to explore Tuscany, then a couple more to Milan.

From Milan, you can reach Paris, London and onwards to your destination in the UK the same day. But you could take a few days over it if you like.

Even if you decide to fly both ways, I’d suggest you fly into Palermo and out of Rome (of vv) and use the train between.

Between Sicily and mainland Italy, the train goes onto the boat, which is quite an experience. The older kids will enjoy that

Just so that you know. If you want to buy seat reservations for your 2 year old child, then you need to get a free children's pass for that child too. Otherwise you need to have that child in your lap. 

Reservations for children have the same price as reservations for adults, except for seat reservations from DB where there is a family offer.