Steve, what a bummer. Like bergs says, looks to be the rings, and oil on the piston crowns is the source of the smoke.
Can't remember if you have had the barrel bored or just replaced the rings. Either way the final stage is to hone the bore. With a rebore, this knocks off the peaks of what is in effect a fine internal thread. If you just replaced the rings, the bore still needs to be roughed up to remove the surface glaze, otherwise the rings will never seal.
So, a few things about piston rings. Assuming you have the right rings for the bore, the top ring has slight variants, and will be either plain, or a barrel profile. These can fit either way up. Some sets have a groove round the top to clear the wear ridge at the top of the worn bore. Another sort you may find is a "tapered periphery ring", where the ring diameter increases from the top to the bottom of the ring. Here the narrower part goes to the top. Very hard to tell by eye. so these rings will be marked "top"
Middle ring, can be plane, barrel, tapered periphery as before, or a type with a circular groove round the outside, like a little hook in profile. This is a Napier ring. fitted with the groove to the bottom.
The side clearance of rings in the piston grooves should be minimal, no more than 1-2 Thou. If the rings slop sideways in the groove, this up and down movement transfers oil into the combustion chamber. No matter how well the rings seal radially, oil will still get by. All the rings should be able to be pushed into their grooves below the piston lands, and be free to move in, out, and rotate round in the groove.
The oil rings are straightforward. Make sure the drain holes in the ring groove are clear. If you still have the old set of rings handy, do a comparison how each fits in the bore, checking the ring gap.
Maybe worth cleaning up the old rings and replacing them as a trial.
Oddly enough some oil is necessary in the bore. If your carb is currently very rich, you get a situation where almost neat petrol covers the bores. Termed "Bore Wash, or Fuel Wash" long term it is detrimental to the pistons and rings, washing any lubricant off the bores.
Really thought you had it sussed, and apologies if you are well experienced with pistons and rings.
Swarfy.