If you're very flexible about your travel dates and times and can book pretty soon, you can get tickets from London to Nice return for just over £100. This is available on Eurostar's French website or as far as I am aware any Eurostar website quoting in euros(
www.eurostar.fr will get you to the French site - I am pretty sure you can still get the normal loyalty points). Buying from Eurostar's UK website (buying in pounds) likely to cost around 10-15% more. I would recommend avoiding the first week of June (British school holidays and Queen's Diamond Jubile) if you want to find one of these tickets as they have virtually all booked up (£130-150 is still available though). You could also try split tickets London - Paris and Paris - Nice on
www.tgv-europe.com but I think that Eurostar's through tickets are likely to be by far the cheapest way of getting to Nice by rail at this stage.
I think that a return from Nice Ville to Torino Porta Nuova is best booked/bought in two stages. It costs 7€ each way from Nice to Ventimiglia (the border station) for a ticket at Tarif Normal on a French TER and then 11€60 on a Regional train (the ordinary Base fare) changing at Cuneo (the fastest option and most frequent option, taking 3h20) to Torino Porta Nuova.
One of the fastest itineraries seems to be:
10.55 Nice Ville - 11.51 Ventimiglia Stazione
12.04 Ventimiglia Stazione - 13.57 Cuneo
14.03 Cuneo - 15.25 Torino Porta Nuova (4h30)
You cost also travel via Savona using an Intercity and buy a Mini (i.e. Advance Purchase) ticket for 9€ each way. Travelling via Genova seems to take longer and is hardly shown by Trenitalia's website. You can investigate further using
www.tgv-europe.com for the French section and
www.trenitalia.com for the Italian section.
I'm pretty certain that the through trains to Genova do not run anymore - all journeys between Nice and Italy now involve a change in Ventimiglia (except when using the Russian Railways' Nice - Moscow sleeper, I think, but I don't even know if you can use this for such a short journey).
A quick search shows that there seems to be confusion about whether railway photography is allowed on Italian railways or not - some people have been stopped by the police for doing so (i.e. guards have called the police). On one website someone had however written to the police who had replied by saying that as long as you have a hand-held camera and do not disturb the working of the railway and its employees or harm railway infrastructure, private photography is permitted. On the other hand, an article from il Corriere della Sera states that Trenitalia's public relations department had told the newspaper that all private photography was forbidden unless one has asked railway officials and has received a specific authorisation to do so.
I have taken photographs at Italian stations before and not been stopped but I do not know what is the norm. You could ask "Posso fotografare?" (pronounced as you would expect) if you are in any doubt.
Private photography is allowed at SNCF stations as long as no flash is used, the railway is not disrupted and the photographer takes the photos from public areas of the station i.e. no trespassing. Photographers are however also known to have been stopped. All photography is forbidden on all RATP services and in all stations (for anyone who does not know, RATP runs the Métro/RER services in Paris; this rule presumably applies to its buses as well although this would be impossible to enforce I think).