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I think you already have most of the answer but there are some additional bits that might help.
Border paste, it's a kind of ready mixed wallpaper adhesive, bit stronger than the usual.
Syringe, you can get a big one for decorating. Use it to squirt wallpaper paste under those edges.
Decorators sponge and clean water in a bucket. Use the sponge to smooth out the paper and make sure you wash all the excess paste off otherwise you might have trouble painting it.Small gaps and damage can be filled with something like Toupret TX110 if you really need to. There's also a tape made by Tesa called crack cover which can be useful in these cases but is not a necessity.
If you are super careful you can get away with decorating afterwards and have it look half decent. Usually needs a couple of coats of paint though, best to use something with a lot of solids in like a specific ceiling paint or high cover paint.
Best method/product for repairing lifting liner paper on a ceiling?
It’s our first house and the budget is tight so would much rather repair than replace.
Excuse the cobwebs and yellow paint, also addressing that today too.