That's a bummer Mike It does sound like a big end bearing (the one between the con rod and the crank shaft), and it's usually on number 4 (the one nearest the gearbox, since that's the furthest from the oil pump, which is right at the front of the engine).
It's not an easy job just to replace the bearings unfortunately, as to get the sump off you need to remove the front subframe. Even then you'll struggle to get the sump to seal afterwards as it uses silicone / RTV rather than a 'regular' rubber gasket, so the surfaces need to be mega clean - not easy when you have oil running down the inside of the block!
As Jardo says, your best bet is to drop a new lump in - should be able to pick one up for less than £200, and you'll probably be able to find a friendly Nutzer who'll help you do it for a few quid. It's not too bad a job, you can do it in a day fairly comfortably on someone's drive (need an engine crane though, of course).
White smoke indicates head gasket usually, which is another reason to drop a new lump in rather than mess around with this one...
As an aside, none of my MX-5 engines have ever used oil, but did you say you did an oil change recently? It's a really good idea to fill it to the max, run the car for a few minutes, then shut it off for ~10 minutes and check the level again. It will usually appear to drop by ~0.5L (the stick is 0.8L from max to min) as the oil filter gets completely filled etc.