Plug and chain for Belfast sink + worktops

Screw it to the draining board if it's wood, or fit a strap around a tap?

You don't - Belfast sinks (AFAIK) didn't have such a hole for a plug chain. They'd have the chain (if any) attached around a tap, or screwed to the wall.

Don't screw it to the worktop. You're using beech AFAIR, and that's going to be a source of staining in the future.

Sounds like B&Q have bundled the wrong sort of plug fittings with your sink. But then what more can you expect from a big orange shed?

Even with a brass screw?

Yes. Obviously oak and iron will stain in no time at all, owing to chemical reaction. Beech and anything (even just the hole) will encourage damp and bacterial or fungal action. As beech is a bit prone to this, I'd be concerned about dark staining around any exposed end grain that you can't keep well oiled.

Personally I wouldn't use beech for a worktop. If I were going to immerse it under varnish, I'd save the money and use melamine. For wood tops I would generally use local oak, otherwise maple or tropicals (ipe)