This may be a dumb question, but I read all the posts on this subject, and I am curious how you came to the conclusion that the transformer was the culprit in the first place? In other words, If this T came out of a charger, there are several other components to consider, and I don't want to make any assumptions. I see that you checked continuity on the windings, but did you connect line voltage on the primary side and then check the output? The secondary appears to be center tapped, looking at the photograph on the other thread. Did you check continuiity and voltage across all the secondary pins? As Nigel said, if you are reading continuity both ways, something should be happening. And, another point, as easy as it is to blame a transformer, and I have done that before, it usually isn't. Granted, a thermal fuse can fail, but if you by-passed it, you are nullifying that possibility. Do you have a voltmeter or are you only using an ohm-meter?
Looking at the photograph, there are three pins on the side the chip is on, I am assuming that is a secondary, but, being that the transformer is rated for 250 line, and it is a black and decker drill charger, I can't imagine it needing more than 110 line, so this might be the primary side, and the factory might be center tapping the primary with 110? Please give us more detail on how it is oriented, and what troubleshooting steps led you to assuming the T was dead. It might be something as simple is the power switch has dirty contacts, and no line is getting to the primary!
Oh, and for Christ's sakes, be carefull with line voltage and a transformer!