Your Oyster cards are good on National Rail trains within greater London only.
To travel by train from London to Nice you will have to take the Eurostar from St Pancras International to Gare du Nord in Paris.
Take the RER D line directly from Gare du Nord to Gare de Lyon (1.60 EUR). Trains run every 15 minutes and the ride takes 8 minutes. Like the Metro, the RER runs underground in central Paris.
Take a TGV from Gare de Lyon to Nice Ville.
Don't fall over when you see the fares for the Eurostar and the TGV! Discount fare tickets are long gone and full fare tickets are pricey. And if you think 3-4 hours from Nottingham to London is exhausting, you're not going to be at all happy about how long it takes to get from London to Nice.
Book your Eurostar tickets at www.eurostar.com.
Book your Paris-Nice tickets at www.tgv-europe.com. Choose Canada as your country of residence. Since ticket machines in French stations only accept credit cards with embedded chips, choose to pick up your ticket at the station. Go to the window and the agent will swipe your card.
Use the German Rail site to get detailed timetables for the whole route and for your route from Nice to Italy. After you get summary timetables for point A to point B, click on the arrow to the left of the departure time(s) of your choice to see the connecting points.
Buy your tickets for Nice-Genova when you get to Nice. Buy your other tickets when you get to Italy.