Train from London to Rome in the daytime

Check out this page from the Man in Seat 61.
http://www.seat61.com/Italy.htm#.Vvyr95X2Zjo

Can't imagine why you want to take a train that far, it's about 14 hrs on the train. You can fly from London to Rome in under 3 hrs (flight time) plus getting to and from airports and waiting time - probably about 6 hrs total - and may even be less expensive. Norwegian Air flies direct from London Gatwick to Rome Fiumicino. Ryanair is cheaper but they fly from London Stanstead to Rome Ciampino which may not be as convenient.

What you would have to do is take the Eurostar from London St Pancras station to Paris Gare du Nord (station). Get across Paris to Gare de Lyon for a train to Geneva. Or from Gare de l'Est to Basel or Zurich. Change at Geneva for a train to Brig and change there. Or at Basel or Zurich. Whichever route you take, when you change you will be onto a train for Milan. Change again in Milan for a fast train to Rome.

If you go in 2 months you can get some cheaper tickets now, but many will have already been sold by now - they go on sale 90 to 120 days out.

You should have plenty of daylight for such a protracted trip in May and June.

According to the Deutsche Bahn route finder, you would need to be on the 5:40 am Eurostar from St Pancras, one hour to change from Gare du Nord to Gare de Lyon in Paris on the RER, TGV from Gare de Lyon to Turin, then Freccia train to Rome Termini arriving at 8:55 pm. That is the only way to do it in daylight.
You can buy all the tickets at www.capitainetrain.com, except for the cross town transfer in Paris, which is 1.80 euro from a vending machine or the ticket window.

Easyjet and Vueling also fly from Gatwick
to Rome. Be sure to check luggage requirements on all budget airlines.

That's my kind of travel! We like to stop for a night somewhere to break up the trip (Paris) but if we have time we will always choose a train trip over flying. Our last trip to London was 8 hours door-to-door flying, none of it any fun at all. I'd rather spend 15 hours on a train. :-) The Man in Seat 61 website and NIgel's fine instructions should get you through.