Why would you change the transformer? In the UK we are used to 110V equipment being 55-0-55 CT-E and not suitable for direct connection to the mains. This machine was designed to be connected directly to the (110V) mains and the transformer is only needed because our voltage is wrong. In a sense the transformer becomes part of the machine to make it work on 230V, not an electrically separated safety source of power. If the transformer were inside the machine (and many machines do have a stepdown transformer built in) would you think of opening up the machine and changing it?
Now it may be that the insulation of the machine is poor and that is a different matter, but it might not be a great idea to sweep an insulation problem under the rug by connecting it to a 55-0-55 CT-E supply and saying 'well the insulation isn't so important now! Indeed a yellow tool transformer with CT-E output might not be suitable, as this requires double-pole fusing wherever fusing is used. If the machine has internal fusing it will probably be in the line only (as it was intended for 110V with earthed neutral) and so the transformer would need modifying to 0-110 and marking as not for tool use, etc. All in all a bit of a bodge rather than an actual solution to inadequate insulation.
The only other reason I can think of for changing the transformer would be if it is auto-wound (although it doesn't look like it from the label) and the machine insulation is only adequate for 110V. With an autotransformer, provided the common terminal is the neutral, no part of the machine would be over 110V to earth; but if the supply polarity is reversed it could be that the machine is slung between 230V and 230-110. The machine would work fine but would require insulation good enough for 230V.
Re PAT testing, the visual is important a) because of deterioration and b) because it was built to different standards and may have class 0 insulation etc. The earth continuity would have to be checked to the transformer case and machine itself if class 1. If the transformer is isolating, then an insulation test from the plug would only test its primary, in which case disconnect the machine from the transformer and test it separately at 250V, and test the transformer primary with one end of the secondary earthed (obviously don't do that if auto!)
To put it in perspective, compared to a lot of the stuff I work with daily, this is neither old nor obscure. I have older pieces of equipment in use in my lab that I wouldn't even think of as old, also lots of 110V gear running on transformers, mostly autotransformers from the 1960s-70s. The oldest kit I routinely subject to 'normal' PAT is 80+ years old and it is mostly fine, provided one appreciates the specifics of its construction and the difference between a non-compliance with current standards and an actual hazard.
Inspect closely, consider the specifics of the transformer winding arrangement, scrutinise the insulation and apply common sense.
