Short answer: ignore so-called bridge cameras. Go for 1’’ inch sensor cameras, or mirrorless systems (Panasonic and Olympus micro four thirds, Sony a6000, etc.) depending on budget and preferences.
Longer answer:
Simplifying somewhat, I’d say the two key things which set expensive cameras apart from cheap point-and-shoot are the sensor and the lenses. 99% of the other ‘characteristics’ companies love to brag about are, IMHO, useless marketing waffle.
Smartphones and compact cameras have very small sensors; mirrorless, DSLR and other more professional cameras have substantially bigger sensors. Look at the Wikipedia page: en.wikipedia.org/wiki/Image_sensor_format ; smartphones and compact cameras tend to have sensors of the 3 smallest sizes listed there. A bigger sensor means a better picture with less noise.
As for lenses, a key characteristic is its aperture, i.e. how much light it lets in. Lenses with bigger apertures are called ‘faster’. The aperture often changes with the focal length, i.e. as you zoom in the aperture tends to decrease. Lenses which can have the same aperture throughout the focal range (i.e. regardless of how much you zoom) exist but are expensive.
Why would you want a big sensor and a fast lens?
(Almost) none of this matters if you are taking a picture under excellent lighting conditions, and are not interested in bokeh (more on this later). With the proper light, you’d be hard pressed to distinguish the photo of a landscape taken with an old smartphone and that taken with a £ 2k camera system. Sure, there are differences, but hardly worth £2k.
However, a good camera can make all the difference with respect to:
*Fast-moving subjects, like toddlers and pets: autofocus speed can be key
*Low light: I have taken plenty of excellent pictures with no flash, using my Sony camera, that would have come out as a black blur with even the latest Iphone or Samsung phone.
*Depth of field: fast lenses give you the option to control how much of the background is in focus. With cheaper lenses the whole background is always in focus; with faster lenses you can control how much of the background you want blurred; this is very nice in portrait photos and is called ‘bokeh’.
Like with smartphones and computers, technology has plateaued somewhat, meaning older cameras can be cheaper but still very valid.
If you want a more compact camera, I’d recommend something with a 1’’ sensor and a fixed, not interchangeable, lens. The Sony RX100 is excellent; it’s the size of a compact camera but with a much bigger sensor. The new versions are considerably more expensive, but not worth it for your kind of use, IMHO.
I find bridge cameras totally useless for your kind of use, because sensor size and lens aperture tend to be the same as those of cheap compact cameras; bridge cameras have longer zooms and many more controls on the body of the camera, but these things don’t make much difference when taking pictures of your kids.
If you’re willing to consider a camera system with replaceable lenses, go for a mirrorless (more compact than non-mirrorless DSLR but with the same sensor size). Panasonic and Olympus make the micro four third systems; Sony makes the alpha which are very good. If you don’t want to buy used, look for older models which can still be bought new.
You can probably buy a Sony a6000, with the standard 16-50mm lens, for £400 to £450. I have that, plus a fast 50mm lens (fixed, no zoom); the same focal length can be reached with the standard lens, but the fixed one is much faster and therefore much better for portraits and low-light photos. I probably took 2/3 of all the photos we have of our daughter with this lens. At some point I might buy a 35mm fast lens, too.
You could ask on the dpreview.com forum for more details; people there will be more familiar with other models and with the latest prices.
Of course making the most of a camera requires learning at least the basics of digital photography; you don’t need to become an expert, but you should at least become familiar with concepts like crop factor, aperture, shutter speed, depth of field, raw files, colour curves, etc. There’s nothing wrong if you don’t want to or don’t have the time to – just be honest with yourself: if you’re not going to learn the basics, buying something like a Sony Rx100, and using it always in automatic mode, is probably a better use of your money, and will still produce better results than a smartphone.
Hope this was useful!
PS A professional photographer would probably balk at the simplifications in my explanation, but the point was to convey the key pieces of information without going into too much detail.
PPS Look into the Adobe Lightroom or Corel Aftershot software to manage your photos.