Thanks for some good advice all round. I suspect the biggest problems I'm facing with this piece are the use of an unfamiliar cutting technique, plus the fact that I've never tried a finishing oil before. It's a silly 'fear of the unkown' thing.
Anyway, I decided to look round our local DIY shed and see what sort of finishing products were available. There was nothing except either varnish or Colron Danish Oil. How I'd have liked to have seen some of the Chestnut products (or even generic shellac)! So I've cut a small section of waste wood and I'm using that as a test piece for the Danish oil. The instructions say to apply a thin coat, leave to dry for six hours, then repeat the process twice. If it comes out to my satisfaction, great! If not, I'll try the aerosol technique Terry describes; I might even drive over to a proper woodwork shop and see what they've got.
Andy, I tried to follow your link but it doesn't seem to be working for me. Could you check it out, please? The tube on my Hegner can be either a blower or a sucker, depending on how it's configured. At present, it's blowing, but if I was working MDF I'd have it hooked up to a vacuum. I find it to be an effective dust clearance system, albeit a bit primitive, and if you've got a more sophisticated setup I'd love to see it.
Gill