Thank you for the photos. It looks like a solid, well-built machine, with a good-size inductor ( choke ), and solid aluminium busbars. It has one large capacitor, which smooths the DC output and stabilises the arc.
The eight voltage readings are obviously wrong, and this implies that the switch and/or the connections to the transformer taps are wrong.

The lowest output voltage - around 15 volts - would be produced by a transformer primary winding with the highest resistance, and this should be connected to the switch at position "1". At each higher step, the resistance of the connected primary tap should decrease. However, these resistance values for the transformer primary will actually be very low, typically 1 Ohm at Pos. 1 and dropping progressively to maybe just 0.2 Ohms at Pos. 8.

The "EN 60947" numbers on the original switch are standards for mains-rated switchgear. Could you quote the part number or type number of the previous switch, and also the Sealey replacement. With those numbers it may be possible to find an operating chart or "truth table" which will show how the switch terminals connect for each position.

There are often two On/Off switches on each of the segments or wafers, and they are opened or closed by cams that are mounted in a staggered pattern on the square operating shaft. In a 1-pole 8-way component, one side of each switch are all linked together to form the "1-pole" , and the other sides are the 8 connections to the transformer tappings. An 8-position switch needs 8 On/Off switches, with a total of 16 terminals, across 4 wafers each with 2 switches. However, from the photos the previous and the current switches have only 12 terminals?
For example, an 8-way:

If the original switch and the new one are correct with only 3 wafers and 6 switches, that means the layout is not a simple 1-pole 8-way. To get 8 voltage steps there must be a "High/Low" type of range selection and then 4 steps in each range. I'm afraid it will not be so easy to sort this out....
An earlier example of this issue:

Our resident rotary cam switch maestro is @Nix2