[Radioroslyn]

Cone is easy to remove. Unsolder wires (2) from voice coil to output transformer. The wires maybe held in place by a small coil/spring. Carefully lift the wire up to release them from this coil with a pair of needle nose pliers. Unscrew the screw in the center and then the screws around the circumference of the speaker's frame. You maybe able the lift the cardboard spacer without it sticking or tearing the cone. If not what you do is take an Xacto or small snap blade knife between the cone and the frame. The object is separate cone from the frame and keeping the cone intact. Once the cone is loose from the frame lift it straight out.

If the spider and voice coil are in good condition I've reused them with a new cone. Generally the old cones are pretty beat after 90yrs. Paper is dry a brittle. You can use some diluted fabric glue around the flex area along the outside edge. This will bolster it up for a while.  

< Do you know if there are any write-ups or YouTube videos on this part of the process? 
There was a fellow here on the Phorum that had one of those style spkrs. He rewound the FC and OPT. The posts are for the Phorum but I'm not sure how to find them. I think it was a couple of yrs ago.  Don't know of any videos.

< I assume that the cone can easily be re-installed after the new coil is wound? 
The cone/vc just screws back in. There is some gluing if you end up replacing the cone. You just have to remember to make sure to aim the vc wires towards the opt.

< not sure what gauge wire should be used, but I used #36AWG on the previous one 
Well I'm not sure either. But this we know the fc current rating is 40ma. So you can use that as a minimum rating to determine gauge. Or use a micrometer to check the size of the wire of the open coil. Then you can figure the ohms per foot. Larger the gauge the longer the wire the bigger the coil.

https://philcoradio.com/phorum/showthread.php?tid=22316