I think about this a LOT. I think being on this island would be inconvenient because of the climate, but I wouldn't have the nerve to get OFF the island and head for a more temperate region of France.
First couple of weeks - explore, assess if I'm really alone, indulge in some frivolity (diamonds, premium wine/food, museums etc.
Gather antibiotics and as big a portable pharmacy as possible. Definitely interested in living a long and healthy life here, so away from cities with vermin etc. and out into the country asap.
Gather substantial number of books, camping equipment, survivalist kit, and load up a van (with as much petrol as possible).
Then find a pretty, period house or small castle in the south of England (i.e. warmer), big enough to be fun, small enough for me to manage alone and not get spooked in (strong "final girl syndrome" fears here) crucially with a pre-existing and well maintained kitchen garden and small farm. Stock with small number of animals and learn about growing my own food!
Would be strongly tempted to find something like this though - www.castleist.com/2-5m-devizes-england-spectacular-grade-i-castle-for-sale - and just pretend that I'm a medieval queen (with convenient access to antibiotics) and roam my estates with my loyal Irish wolfhounds at my heels all day.
I would have a blast and apart from missing my family, I don't think I'd be that lonely to be honest.
This is in the "everbody has mysteriously and unthreateningly vanished" scenario as opposed to the "everybody is lying there dead, including my husband and children" scenario, which is obviously less cheerful.