I see there's been no action here, so I'll add my very amateur thoughts. Hopefully you'll get a response from one of the knowledgeable guys who are active here.

I believe that for two combustion devices (on separate floors) to share a chimney, it's required by the building code that the chimney have flues for each that are physically separate from the other, either with separate channels in the masonry or with separate metal flues that share the space. I think flue sharing is only possible with two appliances side by side, as with a water heater and a furnace sharing a flue using an inverted wye.

Otherwise, the dangerous exhaust of one can backdraft into the other. Such as, the combustion products of the oil boiler in the basement could backdraft into the wood stove and thus into the living space. Not a good thing.

Thanks, Snidely. I guess I'm not clear on the definition of flue. I looked it up, and I'm still not clear. :-)

The flue is the duct that carries the exhaust gas. The chimney is the structure that houses the flue. A chimney may have only one flue. In the case of a brick chimney the bricks are lined with clay tile. The tile form the flue. A chimney may have multiple flues that are not connected with each other. For example an old house that has several fireplaces. Each has its own flue that travel to the roof inside the same chimney.

I am not expert in this area, so I will just share my thoughts. I feel it is dangerous to connect an oil burner to the same flue as the wood stove especially when the furnace is one floor below the stove. Under certain conditions the exhaust of the oil furnace could back flow into the wood stove and the house.

Think of a small raisin box and stick two drinking straws through the open ends. Elongate the raisin box. The raisin box is the chimney, the straws are the flues. Chimneys can be built with separated compartments, or with more modern installations, metal flue ducts are run through the length to keep the exhaust pathways separate. I think each flue should be connected to just one device and then run (without any further connections) to the top.

Thank you both for your help. The house in question is quite old, and the new owners don't know if the chimney has one compartment or two. Perhaps a chimney sweep would be able to tell.

A chimney sweep would absolutely be able to tell. You can also check yourself by climbing up the roof and looking down the top of the chimney. Please be safe.

Callights, it's not my house, but even if it were, you can be certain I wouldn't. attempt it :-)