Let me try to clear up something about these ohms. When you measure the resistance of a transformer winding with an ohmmeter, you will not read either the actual impedance of the winding at its operating (audio) frequencies, or the impedance of the load (speaker) it is intended to drive. So you should not expect to get readings around 8 or 900 ohms.
But the meter readings should be lower than the impedances for which the transformer is designed. In your case, I see a couple of problems.
First, the secondary side resistance is too high. I measured an assortment of small and medium sized audio output transformers, and in all cases the secondary resistance was less than 2 ohms.
Second, if the primary were actually tapped in the center of the winding, then you would get readings like 1-2=13 ohms, 2-3=23 ohms, and 1-3=26 ohms. Since you report that all three measurements are the same, the most likely explanation is that the tap is not in the center, but very close to one end of the winding. The real situation probably is something like this: 1-2=13, 2-3=1, and 1-3=14. Measuring these small resistances accurately is not easy.
Keeping in mind that this is a very long-distance analysis, it appears that you do not have an audio output transformer. The secondary resistance is just too high. You might want to make your measurements again paying careful attention to the Zero adjustment on you meter and the range selection, to make sure you got the numbers right. Are you using an analog or digital meter?
Unlike the Zapper circuit, the syncrometer circuit is much more sensitive to changes in components. The transformer is the most critical component in the circuit. Small changes in construction or materials will greatly affect how, or even IF, the circuit performs. Do you have any more information about this part? Can you send a digital photo, or measure the core?
Besides the US, Europe (especially England), Japan, and Australia have stores which support hobby or amateur electronics, stores where you can buy just one piece of something like a transformer. Is there anything like that where you are. By the way, where are you?