I have a bayonet light socket, in which I have managed to snap off the "teeth" that hold the globe in place.
I'm not comfortable with doing electrical work, so is there something I can do to "fix" it well enough that the bulb will stay in place?
Not really. The "teeth" are an integral part of the design and if weakened won't hold the bulb in place safely.
The best thing you can do is find an electrician who'll change the socket for you. It's a quick job so you shouldn't be charged much, but get 2 or 3 quotes to make sure you find the best.
If you're having problems with bayonet fittings then it might be worth getting it replaced with a screw cap. Even in the UK these are becoming more popular so you shouldn't have any trouble finding bulbs.