With London as your first stop (meaning likely a jetlagged, sleep-deprived arrival for at least some of you), four nights only gives you three fully-usable days. We nearly all have to make compromises to fit our desired destinations in the time we have available; will three days in London allow you to cover your top sightseeing goals? In a large city like London, you're likely to spend some of your time just moving from place to place, even though the Underground is very helpful.
I enjoyed Lyon a lot, and you'll have no trouble filling your time there (which will be roughly 2-1/2 days if you take a morning flight). However, I'd consider dropping Lyon from this trip to allow more time elsewhere. There are aspects of Lyon that are somewhat reminiscent of Paris (to me), so I don't think it provides as great a contrast with the capital as some other French cities would. For a first, short trip, I'd tend to head elsewhere, though I recognize that some others will probably disagree.
I think going all the way to the south coast of France for only 3 nights (2-1/2 days, roughly) is not advisable when you're going to have to head next to Paris. Most folks heading to Nice want to see not only that city but also some of the coastal towns and nearby hill villages, and it takes longer than 2-1/2 days to do that. I understand the desire for a beach day, but to go all the way to southern France for that reason is, I think, very ill-advised. I'd want more than 3 nights to justify the travel time. If you dropped Lyon and could devote perhaps 5 nights to the south, I'd feel better about it.
Marseille isn't the best base for Provence if you're interested in places like Avignon, Arles, Nimes, Uzes, and villages in the Luberon. Marseille is an interesting city, but it doesn't show up on very many first trips to France, and I think there's a reason for that. Most visitors head elsewhere, at least until they've spent a good bit of time in France. What draws you to Marseille as a possible 3-night stop?
The train trip from Avignon, Arles or Marseille back to Paris would be a lot faster than the trip from Nice, as I believe you've already discovered. As for the inbound trip, if you skipped Lyon, you'd want to fly from London to southern France, which would probably mean landing in Nice or Marseille.
I'd suggest a different time allocation:
London 5 nights (or maybe 6)
Nice or a Provencal city: 4 or 5 nights
Paris 5 nights
If you're not wedded to the idea of heading south (where you may be competing with French vacationers for hotel rooms if your trip is in July or August), there are other options closer to Paris that many people enjoy, including the Loire Valley and Normandy. France is a wonderful country with a great variety of destinations; you can hardly go wrong, but it's easy to spend a lot of time sitting on planes, trains and buses if your destinations are not near each other.