After a quick search online, I found a local sewing machine repair person who came to my house the following evening. After a quick assessment, he gave me a quote, took the machine away, and delivered it back to me on Wednesday evening. Easy! I felt very lucky to live in a city, and have such a great service available. I even took the opportunity to ask him for his recommendations for good machines if I wanted to replace mine (a Janome in the £200 region, if you're interested).
My machine - a Singer Inspiration 4205.
The problem with my bobbin winder was apparently caused by something hitting the top of it. Rather than something falling on it, I think this may have happened on one of the rare occasions when I actually put it away in the sideboard, and knocked the winder against the top of the cupboard. In any case, my helpful repairman reckoned that the break was an unusual one, explained how he had fixed it and made the part more sturdy and said that I had a pretty solid machine, despite it being a cheap one.
Have you ever had a problem with your sewing machine? What did you do about it? Are you lucky enough to have a local sewing machine repairman? I think mine might join my short list of essential 'fixers', along with my hairdresser and my local cobbler!