Is there any way of testing whether it is the camera battery or the camera that is at fault if the battery won't charge in the camera (and I don't have a separate charger)?
I have had digital cameras for the past few years and none of the batteries charged in the actual camera,
A battery charger I would think would be your answer, you could then charge your batteries and try them in the camera.
Or try other batteries maybe, and see if they would work.
I use Nikon EN-EL3e batteries, which are flat oblong batteries., Did you not get a charger with the camera? I always bought a spare battery for mine, always handy for a back-up.
No separate charger with the battery- charges in the camera and I don't use it enough to warrant purchasing a charger. Leaving it to charge in the camera suits my needs.
You don't say how old the battery is; they do not last forever, so I would suggest taking it along to a camera shop with the camera and get a replacement. The shop may even be able to check it for you.