Matches, candles, batteries, battery or wind up radio, torches, pack of cards, flask, hot water bottles.
Powdered milk, sugar, hot chocolate powder, teabags, instant coffee (ok, maybe grounds if you have a plunger but instant uses less water and is easier in adverse conditions).
Pasta, rice, noodles, gnocchi (long life packs), powdered mash - enough carbs for dinners for say 4 days of the type that your family likes.
Part baked bread is a good idea, or tortilla wraps, long life bagels or pitta breads, crackers etc. I also bake, so have flour and other necessary ingredients available and can make bread or scones, some buns or biscuits. Or whip up some pancakes.
Tins of things like baked beans or spaghetti hoops, tinned or packet soups, tuna, sardines, etc. Easy to heat or just eat cold things that you would have for lunches or snacky meals.
A couple of jars of pasta or curry sauces, maybe a pour-over sauce for meat in the oven, etc. Again, you're trying to have about 4 days' worth of dinners that can be cooked easily if you can't get fresh and also potentially have lost your main cooking facilities (especially if electric).
Dried fruit - it's good to snack on and has plenty of calories but lasts a very long time. Raisins or sultanas are nice as snacks or in scones, apricots and mango are both nice to snack on, and Lidl do a bag of mixed fruits including pear halves that is nice too.
Some biscuits, and definitely some chocolate and some boiled sweets. As much to replace energy fast if needed, and comfort foods. And can be eaten cold. A few cereal or energy type bars. A box of cereal. Jam, marmalade, nutella or honey for bread/toast (according to family preference).
Cans of soft drinks, a bottle of squash to dilute (can use hot water), yes wine/beer, a few bottles of water (and a few containers to put water into if frozen pipes are threatened).
Toilet rolls and washup liquid (you may already washup daily, but some I know only ever use the dishwasher).
First aid supplies - band aids etc, pain killers, cough/cold/flu rememdies, and ones for upset tummies/guts. And a few spare days of any vital prescriptions the family uses (or just refill when bad weather is threatened - there's usually enough warning).
I also have plenty of hats, gloves and scarves around the place (I have multiple pairs for DD as once it snows, she spends a fair bit of time outdoors and comes in with wet gloves, but wants to go out again as soon as she's warmed up). I also have wellies and rubber trousers for her generally, and we all have decent outdoor gear, that gets well used in snow (like you, near the sea so it's rare, but even though suburbia, lots of locals here were badly caught out a few years ago!). And fleecy blankets to throw over ourselves.
We also keep plenty of board games, books and dvds etc, which can occupy us whether we have power or not (we have a DVD player for the car that can play maybe 2 movies when fully charged). (Charge up all chargeable devices when snow or bad weather potentially threatening power supplies is forecast).
And we do also have a closed stove to have a fire in - do you have alternative heating and cooking options? Make sure you have fuel for the fire. Do you have a gas cooker or rings already, or a solid fuel range? Or a camping stove or the 2 ring cookers that they used to use in bedsits? We have a dual fuel cooker (gas rings, electric oven) - but we also have the option of wrapping up warm and cooking on the BBQ outdoors if we really have to. (And we have a traditional BBQ on the Green in front of our street on the first snowy Saturday in any winter - we have mad French neighbours who instigated it, and about 3 family BBQs come out and everyone brings what they have in their fridges, followed by hot wine!!).