Continuity testing: Blue to mains is nothing -- no buzz. Blue to black gives 039 buzz. Blue to the other two gives 043 buzz. Everything else gives 000 buzz.
What does that mean?
It means that "continuity" is
not the appropriate range to use for these tests, as the primary winding may have a higher resistance than can be tested with continuity, & still be OK.
For instance, I just checked my similar (230v) clock radio, & found the primary is 334 ohms.
(If you have 127 v Mains, it will probably be about half that)
Continuity range will see that as "open", so use the "ohms" range instead.
*From your first photo, it is quite obvious that the brown & blue wires are the primary, as they come out to the Mains power cord.
Those colours are pretty much the standard for Active ("hot") & Neutral ("cold"), respectively, everywhere except North America, who like to go their own way!
If you hang across the pins of the plug on the other end of that cord, you will be looking at the transformer primary.
*I don't know why so many people on this forum are so enamoured with "continuity", which is mainly meant for testing large numbers of cable connections in awkward spots where it is hard to look at the DMM display.