London to Nice is 580 nautical miles and takes under two hours in a light jet, which makes it one of the most-flown private routes in Europe. Every aircraft class can do it non-stop, so the choice is about cabin and cost rather than capability — and a light jet is genuinely enough for four people.
Private jet charter from London to Nice starts from $10,000 on the smallest suitable aircraft. That is an indicative starting price, not a quote: the figure for your trip depends on which aircraft are available, where they are positioned and the date you fly.