[MrFixr55]

The rectifier winding may run hi due to no load, but it seems that the replacement transformer HV secondary voltage is not far out of range considering your no-load measurement 0f 690V vs the 700V rating for the Hammond.

Per the RCA Receiving Tube Manual, the output of a type 80 or 5Y3 tube with an AC input of 350V per plate (what you get with a secondary of 700V with a center tap) varies between 300V at the input of the filter for a 20mA load and 260V for an 80mA load.  Caps for the Radiola 18 PS are only 1.0uF, so I think that the choke values are rather big.  The load on this power supply not including the caps is rather low.  the biggest current draw of an individual tube is the 71A Power Output tube at 20mA at 180V.  At 135V each 26 draws 5.5mA for a total of 22A for all the 26s.  The 27 is used as a grid leak detector, therefore current draw is negligible. Therefore, total B+ current draw for the receiver chassis is about 42mA.  (You can actually touch the tubes in this set without burning your hands.  Measuring the inductance, resistance and calculating the voltage drop may be a good class exercise. (Don't ask me how to calculate it, My Bachelor's Degree was in Life Sciences

Attached below is the info for the Type 80 from the RCA Receiving Tube Manual Version RC10, published in 1930. 

RCA 80 Rect Specs fr RC10 Manual.pdf Size: 429.82 KB  Downloads: 23

Please note that the current curves below this diagram are for RCA 81s, not the 80s.  the graphs above the diagram pertain to the 80
The "typical" full wave power supply circuit is similar to the Radiola 17 which used a voltage divider for the needed voltages. The Radiola 18 differs in that tube current will have a greater influence on plate voltage (It was common for the voltage divider resistor on a Radiola 17 to open.)

Hope this helps, especially for your teaching efforts.