Not sure I have an answer for you, but here is my story. I have a B17 on one bike. Not sure how old it is. It might be 5-6 years old, when I put the saddle on the bike. Or I might have bought the saddle 10 or so years ago and just never used it until 5 years ago. Not sure. But I went to the basement and tried some wrenches on the nut. 1/2" was too small. 9/16" was too large. 13mm was too small. 14mm fit but was a bit loose. The nut on my B17 seems to need a 17/32" wrench. Or a 13.5mm wrench. Neither of these sizes of wrenches exists in the world. Unless Brooks had custom sized wrenches made for their saddles. If you had a stubby 14mm wrench, it would work. Longer wrenches hit the top tube. Or if you took a 14mm cone wrench and ground the sides off, it would work really well. 14mm is a bit loose but will turn the nut fine.
After writing all this, I wondered if I had the official Brooks wrench. So I went back to the basement and looked through the bike tools toolbox and found the short chrome official Brooks wrench. It has 12 corners in it and fits around the nut, does not slide on two sides like a normal wrench. Its a custom wrench obviously. It fit my B17. I measured it and the way I measured it, it came to 9/16" inside the wrench jaws. But as I said, 9/16" wrench was too loose on the nut.