Welcome. It used to be at Nice when we went there was no way to check in more than 2 hours before your flight.

I would suggest leaving the luggage at the hotel or the airport go back to the city and enjoy the day with the children and get a taxi or bus back to the airport later. Fortunately the airport is not too far from the city.

You didn't say if its a BA holiday package or not, if it is you can contact BA holidays duty office and get them to change the airport transfer time to a later time to match yiur flight.

As others have said if the flight is cancelled you will be put up in a hotel at BAs expense; this will include meals (in this case Breakfast I guess) but not alcohol. BA will also pay reasonable transfer costs to/from a hotel. Worth having a looking to see what's close to the airport, sometimes BA will organise it, other times you might have to fend for yourself Normally BA will pay £200 a night per room (you might ger more if you need two rooms for your family)

Depending on tge timing of the cancellation you might be entitled to eu261 compensation, the group of experts here can help you understand that too.