Hi Eric
I can’t help on flights, but would argue that 10 days for Rome and Florence would give you plenty to do and see without wasting precious holiday time spent getting to/from another base. That’s not to say you can’t incorporate a wine touring day out of Florence, or a day trip to Frascati from Rome, or day trips to places such as Lucca, Carrara, Bologna, Siena, San Gimignano etc. FWIW when planning I like to have day trip ‘options’ prepared, but usually not booked. Thus when we are in situ, we can make a call on how we feel / what we fancy. We might do one every 2-3 days or none at all.
The trains are good in Italy and trading up to 1st class (booking in advance) can be excellent value and a good experience in its own right.
You can also get the train between Paris and London, going through the channel tunnel (aka the chunnel). This gets you closer to the centre at both ends, so should be easier. Given you are staying in both cities, I would recommend that rather than flying.
Whilst you could also train from Italy to Paris, that’s a much longer journey (~8 hours from Torino to Paris IIRC), so based on your destinations, that would eat up much more time than via airports. Still, it’s an experience I do want to do, as the ability to move around, take your own picnic and wine onboard, plus easier views etc. means such journeys can be a joy rather than a chore.
If going for flights to the UK, watch out for baggage weight restrictions /charges on budget carriers (e.g. Ryanair & EasyJet). Both are typically unpleasant experiences, but when convenient, I’ll still use them (e.g. out of the way airports might be ideal for where I’m going). Note also that Stansted is a fair way from London (though Heathrow & Gatwick aren’t exactly close)
In terms of direction, I’d be tempted to reverse the order - London will feel the most familiar in terms of language spoken/written, so might be the best location when jet-lagged / tired from the flight. From there to Paris and then onto the 1st Italian city and then the 2nd one (which you’ll fly out of).