After being able to make many trips to Europe over many years, I appreciate more and more spending more time in fewer places. Having said that, there are still folks, especially if they've never been (and there's no guarantee that anyone will ever get another chance - things can happen), who can justify an "If it's Tuesday this must be Belgium" kind of adventure, seeing more places but with less time in each place. And Nice, France really is nice!
One of my earlier trips included England (connecting through London on my way to France), France (Provence and French Riviera, and Italy (Cinque Terra on the northwest coast, plus Venice), but that was over 3 weeks, not just 10 days. This was also almost 20 years ago, before the super-fast trains in France and the budget airlines like EasyJet made it possible to get from one place to another a lot faster and cheaper today than in the past.
So, if you were just getting a taste of a few must-see highlights in each location, you could hit London for a 2-3 days, fly or Eurostar train to Paris for 2-3 couple days, fly or take the TGV super train to Nice over a couple days, then fly or train across northern Italy to Venice. I haven't been to Venice since my third trip there in 2004, and my understanding is that cruise ships now deposit thousands of people a day, so it's a lot more crowded than 15 years ago, but it's always been a magical place - for centuries - and see it when and how you can, while it's still there. Once the cruise passengers go back to their boats for the evening, you'll have the city, canals, and Piazza San Marco with just yourselves, locals, and the overnight visitors.
As mentioned earlier, flying Open jaw, into one place and home from another, maximizes your time when covering a lot of ground. Either way, you'll be in great places, but moving around can be more hectic, and going to fewer places can potentially feel like you missed seeing something when you were so close . . . or it's the inspiration to come back on another trip to see what you didn't see this time! Happy planning!