"White spirit" (UK) is the same as "Mineral Spirits" (US), usually sold in the UK as a decorator's brush cleaner and general purpose organic solvent. It's a mixture of aliphatic and alicyclic C7 to C12 hydrocarbons, with a maximum content of 25% aromatic hydrocarbons (but, of course, everybody knew that anyway!).
Mineral oil is probably most easily obtained in the UK as baby oil - any good chemist, or (as some kind people informed me on this forum) at about half the price in most supermarkets. It's very similar to 3-in-1 household oil, but significantly cheaper. If anybody has access to thin hydraulic oils, they're much the same, as are thin lubricating oils, but they tend not to be as refined as that sold for pharmaceutical use, and have the potential to be mildly carcinogenic with frequent contact.
Not sure what's in 'Honerite'; it's a proprietary concoction sold by Norton (I think). Whether it's any more effective than mineral oil (baby oil) is moot.
I think some people have used machinist's soluble oils (which aren't really oils, but coolants). They may work, but are not intended to lubricate; the newer formulations don't suffer quite the biodegredation and stink of those familiar to engineering apprentices thirty years ago. Some people may use machinist's cutting oils, but be careful with these, as they have a reputation for being mildly carcinogenic if used frequently. Diesel is probably best avoided, too, for the same reason; it can also cause dermatitis.
All sorts of thin, slightly oily substances will work (olive oil aka 'sweet oil' used to be the oil of choice in the days of non-porous stones, but best avoided on modern man-made porous stones), but some are more pleasantto use than others, and possibly safer health-wise long term.
