[Arran]

I think that it needs to be mentioned that the BOL machines are all at least 83 years old, as such the paper capacitors, as well as a good number of the resistors, are going to need replacement for the units to work reliably, if they still work at all. If an auto transformer has continuity then it's probably fine, barring something like shorted turns in the winding, so I would say say that the problem is elsewhere. The problem is either ahead of the autotransformer, or after it, if there is a paper capacitor, or resistors after it, check the resistors, and replace the capacitor, or capacitors. A bad tube is possible, I'm not familiar with how these BOL sets are wired but if the pre-amp tube sits on standby, such as being operated with the heater on, but no B+ for the anode, that can cause failure. Flakey tube sockets are another curse of sets with locktal tubes, which is why they often need the socket contacts and tube pins cleaned.
Regards
Arran