My wife suggested I should sign up to Mumsnet just to contribute my answer to this question, so I have! Background: my DD is 15 and has been into photography for several years. It's her big thing.
At any given budget, you could go for a "compact" (one do-it-all lens, pocket sized), a "bridge" (one lens with a big zoom = bigger camera), or (once you're in the £500+ range) an interchangeable lens camera. If you want good night-time pictures, you need two things: a large sensor (so more light hits it!) and a high-quality ("fast" in the jargon) lens.
Most compact and bridge cameras have the same size sensor as phone cameras (1/2.3") with a few exceptions. But they have bigger and better lenses, so still show an improvement over phones. If you spend more (£500ish), you can get compacts with bigger sensors (and better low-light pictures), either 1" or 4/3" (known as Micro Four Thirds).
Bridge cameras are about the same price as a good compact, and generally have a lot more buttons and dials, so they're a good introduction to stepping up to Interchangeable Lens Cameras; but are bulkier, and many of them still have the small phone-sized sensors.
If your DD is interested in portraits and landscapes and "general photography" then a good compact might fit the bill. If like my DD she wants to branch out into wildlife photography, then the size of the zoom is pretty critical.
Interchangeable lens cameras all have larger sensors (either Micro Four Thirds, or something called APSC which is even larger, or Full Frame which is larger again). But they get expensive quickly once you add in lenses. It depends a lot on whether you'll want a big zoom for wildlife (that gets expensive AND heavy). There's a separate discussion about "DSLR" versus "Mirrorless" but this is already a wall-of-text!
What we did was start with a £100 camera. When it became a really serious hobby, we stepped up to a £400; and now have an interchangeable-lens system. (I inherit her old cameras! and am starting to get into photography myself now.)
As for brands, we've been VERY happy with Panasonic cameras.
I'd be happy to answer questions or give specific recommendations if you can talk about budget and type of photography! Good luck... it's a great hobby.