Thanks Paul.
In my case, it was oil in the exhaust silencer (muffler). I took it off, started the engine and ran it for a couple of minutes. No smoke. So, I flushed the silencer with petrol, then hot soapy water, thoroughly dried it off and, seeing as it was off the engine, prepped it and applied a coat of high temperature paint. I refitted it and started the engine. No smoke!
What seems to have happened is that when I tipped the mower on its side by about 45° (exhaust side down) to remove the engine as part of the restoration process, oil flowed into the silencer (possibly via the open exhaust valve?), even though I had drained the engine via the sump plug. From now on, I will only tip the engine backwards (spark plug up) as oil can't get into the exhaust or inlet/carb that scenario.
One note of caution regarding running an engine with high oil level (e.g. to burn off the excess) - A mechanic friend some years ago told me that high oil levels can mean that the crank bearing journals and conrods are splashing about in the oil, and cause lots of air bubbles in the oil. These are then sucked up by the oil pump pickup pipe and with prolonged use, can cause engine wear damage. He was talking about an engineer's report on a car engine failure, so I am not sure how that relates to single cylinder vertical mower engines, but they do have a sump where the crankshaft spins, so it is probably true for them too. I would recommend lowering oil level by sucking some out through the filler hole, with a pump or even a thin turkey baster, rather than running the engine for quite a while to burn off the excess.