OK. Well, we may be starting to make some sense here, but you may not have the transformer that you need.
The old transformer--It has 1 set of primary windings (i.e. 2 cables on the primary side)? How many sets of secondary windings does it have (i.e. how many output taps)?
So, if a transformer has multiple output taps, then the most likely scenario is that the output is putting out multiple voltages for different parts of the circuit, or to different PCBs. BUT, that may not be guaranteed. Some transformers have multiple output taps simply to allow flexibility on how they are wired. But, in your case I am beginning to suspect that you have multiple secondaries that are supplying multiple output voltages (like maybe 12V, 5V, 3V or something similar). Does that seem to be true?
The problem is that your NEW transformer has a single output tap that supplies 12V. You see the problem here?
But, let's make an assumption that you have the proper transformer (and that may be a very big assumption at this point).
If you have located the mains line inside the chassis and have connected that to the primaries on the new transformer, then you have at least connected the input of the transformer to the correct connection. Hopefully this isn't directly connected to the mains line, but rather connected to the mains through the power switch and the fuse.
Now, let's focus on the output. The output of a transformer should be going to your bridge rectifier. A rectifier is a set of diodes that will convert an AC signal into a DC signal. Typically the output of the rectifier will be going over to some power filter capacitors, and the filter caps are typically the largest (electrolytic) caps in the entire chassis.
Can you locate your bridge rectifier in the Power Supply? If you can find that, then you should be able to trace that back on the PCB and see exactly where that connects to the output(s) of the transformer. Since your old transformer has multiple output taps, you may have more than 1 bridge rectifier.
Now, what does a bridge rectifier look like? It will either be 4 discreet diodes connected in a diamond pattern, or it will be a single IC.
This schematic should make it clear as to how the transformer connects to the rectifier:
I'm guessing here, but I would think that you would more likely have an IC rectifier instead of a discrete rectifier. An IC rectifier will likely look like one of these pics. Dunno which one you may have, and if it is Surface Mount then it may be tough to find.
https://www.google.com/search?q=discrete+bridge+rectifier&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjt4P7v-fvTAhXrwFQKHX1dDdIQ_AUICygC&biw=1280&bih=827
This is a general schematic of a power supply, this should help to visualize everything I'm trying to tell you. Can you study this schematic and find the corresponding components in your gear?