A difficult situation Kath. Late 60s.......... possibly flue blocks (special blocks with a hole up the middle) or, more likely, ordinary concrete blocks with clay liners.
Either way, it is quite possible that there is a break in their flue structure that's letting a tiny amount of smoke through the separating wall.
It would be all too easy to accuse your neighbours unjustly and create friction, so I think you may have to talk to them tactfully. Mention the problem to them and ask if they had the flue re-lined before installing the burner.
In a semi or terrace situation it is always a good idea to re-line. No one ever knows the condition of their chimney flue.
Worst case ...... they refuse to co-operate. Environmental Health would take this seriously. A depressingly high number of people are killed or taken ill every year with the effects of combustion gases.
With these gases, what you actually smell is just the tip of the iceberg.
Do have a chat to them.