Hi Bill,
I've replaced a number of Brooks saddle covers on new powdercoated rails when the old (chrome) rails broke. This meant re-riveting, something you won't have to do to simply replace the tension bolt.
Mine were all on B.17s rather than Flyers, but I don't see that making much difference.
To replace the pin, you'll need the saddle off the bike and free of the seatpost and clamp. I prefer to invert the saddle and place the cover on a soft towel to prevent marring the leather cover.
Where the old pin snapped, you're halfway there; the cover and nosepiece should slide free of the remaining bolt end with no problem. DON'T PRY ON THE LEATHER! You may need to ease the steel nosepiece gently over the tension bolt end using a broad, straight-tipped (slotted) screwdriver. I use a small titanium pry bar, but the screwdriver will work. Be sure to note where the nut is on the remaining tension bolt shaft. This will be your reference for indexing the tension on the new bolt.
To replace it, you'll need the nose free of the old bolt end,so remove what's left of the end , the nut, and the shaft. Insert the tension bolt fully in the rails, the nut screwed all the way forward to minimize tension. Lower the rails into the inverted cover so the tension bolt end is on the cusp of the nosepiece.
With the bolt threaded in all the way and no tension on the cover, it should be fairly easy to sort of >>pop!<< the edge of the metal nosepiece over the bolt shoulder. Once again, DON'T PRY ON THE LEATHER! There is a sort of knack to it, but it is something that really only becomes obvious when you are prying. I can say it takes surprisingly little effort once you get it right.
Once the parts are mated, then simply retension to prior levels and you're done.
Yeah, I'd call it a pretty straightforward operation. With practice, I found bolt swaps/installations could be easily done in less than a minute.
All the best,
Dan.