Just to be crystal clear here ,
you removed the plug from the mains I assume , then measured across the live an neutral legs of the IEC input socket on the equipment for continuity within the unit itself ? It half seems from what you said above you checked the live mains outlet at the wall for continuity , in which case you probably would have blown the crap out of your multimeter . If theres even a slight chance you dont know what your doing , DONT TOUCH LIVE MAINS POWER WITH YOUR MULTIMETER .
Lets assume you and your multimeter are still alive , and you have established continuity across the transformer primary , Now clean off the hot glue and disconnect the white plug with the grey orange and yellow wires to the circuit board from the transformer secondary , check each pair in turn for continuity , also check for shorts between yellow grey and orange windings . your still completely disconnected from mains power at this stage .
Assuming all the above checks are good , now apply power to the unit and check each secondary pair( Grey orange and yellow)on the transformer on the AC volts range on the multimeter , bear in mind you may still be dealing with dangerous voltages ,so dont touch anything with your bare hands and dont allow the probes to short together while measuring.
You should read a different AC voltage on each of the three windings , one for the 5 volt rail , one for the 48 volt supply and one for the +/- 15 volts supply . If thats all present and correct you know your transformer is working , the problem is further on in the circuit .