I vote for pre-1978 Perthshire Paperweights PP1 (if it is 3 inch diameter) or PP2 if it is a medium size (2.5 inch diameter)
Fits all of the basic description: No signature cane (= pre-1978); fire polished base; thirteen spokes (13 / 14 were regular in that design [although other numbers did occur]); 1-1-2-2 pattern of canes between the twists. Also, the canes are very crisp and tidy.