What foods can last 3-4 days without refrigeration?

First of all, depending on the area you're going to be camping in, is a camp stove out of the question? I don't know what's readily available to you in England, but there's quite a range of different options out there from little folding metal things that fit in your pocket that use solid fuel tablets up to briefcase sized ones that are basically like a regular kitchen stove that use 1lb propane tanks (or larger tanks with an adapter) and basically every form factor in between using just about any kind of fuel imaginable. You can even make a small stove out of some soda cans that burns denatured alcohol. You can probably have most of them delivered to you from Amazon before you leave for your trip, and assuming prices are similar in the UK to the US, there's options out there that will only cost you about £10-£20 plus a couple more bucks for fuel

Some of them are practically like cooking on a blowtorch so the flame is too concentrated and intense to do much besides boiling water, but even that opens up your cooking options a lot. Being able to heat up your food or make some coffee/tea/hot cocoa can be a huge quality of life improvement. And having a method to boil water to sterilize it in an emergency is always a good idea.

Usually, at least in the US, camp stoves are exempted from fire bans if that's what you're up against, look into your local laws about that.

Beyond that, your options depend a bit on how comfortable you are ignoring "refrigerate after opening" warnings on labels. A lot of things will say they should be refrigerated but would probably be fine for a few days as long as they're stored with a little care (container with a tight lid, wrapped up careful in some plastic wrap, foil, wax paper, etc. kept clean, dry, kept in a shady place out of the heat of the sun, etc) and depending on the type of camping you're doing, if you can bring a cooler full of ice you can bring just about anything you would normally keep in a fridge, some things can be kept cool by submersing them in a cool stream. Anything that's very sugary, salty, packed in brine or oil will probably last at least a day or two after opening. Many of those types of foods originated as ways to preserve stuff anyway.

But assuming you for whatever reason absolutely cannot have any type of cooking appliance and no access to any way to keep your food cool- Most fruits and veggies will last fine for a couple days unrefrigerated. Breads should be fine. Jams, and jellies, pickles, and other types of preserves are probably fine. A lot of smoked/dried/cured meats should be fine (jerky is a classic choice, salamis, summer sausage, etc. should also work but try to get a whole one, not pre-sliced, country ham if that's available across the pond would probably do the trick, maybe prosciutto) Dried fruits, nuts, harder cheeses. In general you can take a lot of inspiration from a charcuterie board. A lot of "just boil water" kinds of foods don't necessarilyneed the water to be boiled, it just makes it go a lot faster, you could do some overnight oats, instant noodles, etc.

As for your ideas, falafel and fritters are probably fine, but will depend on the exact ingredients and recipe you use. Calzones are probably fine, but I'd keep away from using too many wet ingredients in them, keep a separate jar/can of sauce to serve them with, don't load them up with too many veggies and such, stick to cheese and maybe cured meats like pepperoni for the filling. I had to look up pasties because we don't really have them in the US, but I'd say largely the same as the calzones, probably fine but be smart about what you put in them. At any rate, if you don't have a plan to keep them cool, I'd say you probably want to plan on eating them the first or second day, they might start getting a bit questionable after a couple days.