Indeed, this was in policy. Frankly, cancellations due to the airline with no possible rerouting for a couple of hours put you in a position of strength, and whether this is on an avios ticket or not makes no difference. With such late notice, you would have been entitled to EU261 compensation and full refund so it is in the airline's interest to find an accommodating solution instead, and as c-w-s says the BA policy is 300 miles (some airlines like QR have more generous radius).

I really wouldn't recommend LYS personally: trains for Nice to Lyon take a long time (over 4h30) and at this time of the year easily booked up (you need a booking on the TGV) and expensive when available (at least €50-100) to which you'd need to add cost of hotel night (unless you feel like travelling overnight which is tiring) plus additional train from Lyon to LYS airport.

Milan is a bit longer (4h45 nonstop) and there are fewer of them (only 3 a day) but often a bit cheaper. If you have a rental car, however, it is a much better solution as it is only 3 hours driving whilst Lyon is over 5hrs. Again, hotel could add to the cost.

Marseille should be easy but is a pain in the back by public transport. When you go from Nice to Paris by train, it takes almost the same time to cover the Nice-Marseille portion (2h30 for about 180km) than the Marseille Paris portion (3hrs for over 600km!). I won't bore you with the historical and political details of the why. You then again need to change into a bus or train to Marseille, and as MRS is a significantly smaller airport than NCE, you have direct coaches from Marseille to Nice airport but not the other way round. Again, if you have a car, however, this would definitely be the best solution. You could leave Nice around 4am and make the early flight from Marseille although this is not a particularly pleasant motorway around Marseille.

Personally, I would have just tried to fly NCE-LHR the evening before and spent a night in a hotel at Heathrow. It would have been more relaxing and cheaper than all of the alternatives I can think of bar pre-paid last night in Nice, car that you can return where you want, and drive at 4am from Nice to MRS to fly from there.