Pretty much what brunos says. If you are on the 13.20 LCY, then I think that your window of luck is if you arrive when they are still taking luggage for the Gatwick or even the 11.45 LHR but nobody on those flights is at the check in counters and the too-early-347 (the 13.45 to LHR) pax haven't started coming yet, especially if one of the check in staff recognise you as one of the regulars. Then, they might take your bag despite the flight not being open yet, especially as LCY is a smaller flight with more point to point and thus a lot less people checking luggage.

Conversely, I'd say that the worst chance of leeway of all is people coming early for the 347. Precisely, check in for the 11.45 and the LCY would still be open then, and there are just way too many 347 pax who show up more than 2 hours before check in with tonnes of luggage, so staff will invariably make them wait on the side.

All in all, there are four things to bear in mind:

1) As everywhere else except the London hubs nowadays, check in staff are sub-contracted, so they are not there to do overtime unpaid to their company by BA which contracts exactly 120 minutes of check in time for each flight, let alone to spend extra time at the check in counters;

2) BA has significant density on Nice routes - up to 16 daily flights some days, and they need to make it all work and check everyone in on time so if you start allowing the early pax checking in before they are allowed, you may well have last minute passengers for the flights about to close who are stuck in the queue thinking the (too early) people in front of them have the right to go before them (see my recent "please, stop queuing" thread for details), and those last minute pax may then miss their flight whilst they were on time or you would delay the flights to let them on which costs money and aggravation to all involved;

3) Some people are annoying: so you can't just say yes to one pax you shouldn't be checking in and then say no to the next who has seen you check in the previous guy because you never know if it is someone who might compare, insist, complain, and basically just make the check in agents' life miserable just because they tried to be nice to someone but not specifically him//her;

4) In some places you could get away by saying you can just be more lenient to priority pax (Club and GCH/SCH/OWE/OWS) but at NCE, those frequently represent half or more or your total pax, so there is no way you could use that criterion. Moreover, there are many long haul J and F pax who may be checking in 3 or 4 bags each so your leniency can results in several minutes of check in procedures for just one passenger.

So technically, to be lucky you need to arrive at the time when the check in counters are actually open for the late passengers of a previous flight but at the same time when there is no one around which legitimately need to be checked in nor could see you and start the whole shenanigans of effectively opening check in for your flight 2h30 before flight time instead of 2h. From that point of view, 1) and 2)+3) are a bit of a catch 22 because low season makes it more likely that if you arrive at a time when the check in counters are closed if you are early, whilst high season makes it more likely that whatever time you arrive there will already be people queuing creating problems 2) and 3).

There is an obvious - and equally obviously unrealistic - solution: till a few years ago, BA had their own ground operations at NCE and then everything was great. Since they stopped having their own ground staff at outstations, however, the problems I describe above emerged and quite frankly, as much as I'd love the airline to re-own their own check in and reservation process, that's just not going to happen. That is what ultimately makes early check in a strictly enforced no-no in the vast majority of cases unlike what BA can offer (at least for priority passengers) at LHR where they have their own check in staff.