Laura,
I usually wouldn't suggest that many locations with such a short trip, however I do believe it would be possible with careful planning. The disadvantage is that you'd only have time for a very quick look at each place. You'll have to decide if you want a somewhat "relaxing" holiday, or a "blitz"?
With the places you mentioned and such a short time frame, using open-jaw flights would be essential (into London, out of Rome), even if these do cost a bit more.
As you'll have to allow a day on either side for travel to and from Europe, could you indicate whether it might be possible to get a few extra days of holiday time?
Are you absolutely "set" on visiting those locations? If so, I'll try to offer a few suggestions.
Cheers!