Thanks, Eddie to really help me out. Much appreciated. I checked the small potted transformer again by setting the meter at different Ohms setting - 0-200, 0-2000 & 0-20K.From your measurement of the "open-circuit" primary of the small potted transformer, yes, you have to replace that component. With a new transformer there will be power on the PCB, the trigger signal for the Triac will be available as normal through the original torch switch, and hopefully the welder will be fully functional again!
( Regarding the "open circuit" : I may have been low with my estimate of 200 Ohms for the primary resistance. If it is actually 300 Ohms, that would not show on a meter set to the 0-200 range, so maybe you need to test again on the 0-2K Ohms range. Hopefully you already did that.)
To remove the transformer from the PCB, the normal method would be to use a solder sucker, or solder wick. With only 4 pins, it should be possible to just heat the joint at each pin in sequence, whilst levering it carefully upwards.
To find a generic replacement transformer, search for "12v PCB transformer", or "potted" or "encapsulated". E.g.:
Single primary. Ambient temperature: ± 40°C - 70°C. Temperature Class B (130°C). Power: 1.5VA 50/60Hz. Dual secondary windings.www.ebay.co.uk
At 0-20K the primary reading shows 2.89 ohms but nothing at 0-200 or 0-2000. Does this change your advice to replace the transformer? Also, thanks for clarifying to MoreWellie, both of your advice have been brilliant.