No, the 31 jo kata and all the other ones you mention came from Saito, authorized by Kissomaru Ueshiba, of course. Saito took years trying to get all the jo basic moves into one kata. and he finally came up with the one you see.
My school was in Asuma Air Basse, under Isoyama, uchideski at Iwama since he was 12, the youngest 6th dan aikika ever (25) So, we were learning Iwama style as early as 1963.
.We were technically under Saito, because o'sensei wanted Saito to teach Isoyama, but we never saw Saito, unless we went to Hombu, Saito couldn't come to Asuma, he didn't have a military pass.
We did a lot of techniques different than Saito. Techniques that Isoyama thought were more like the way o'sensei wanted them.done.
Nishio was older, eleven years older than Saito, and his techniques are great. Very powerful. He learned them all from o'sensei, of course.
Where and when and how Nishio spread his techniques to the west, I don't know, but I think they were mostly from seminars..
I feel compelled to comment on this. I don't think Nishio sensei got his weapons work directly from O sensei. I think each and every aikido sensei that uses weapons does so based on their own reflection.
In Denmark one of the major branches of Aikido is primarily (almost exclusively) influenced by Nishio sensei, and last weekend we had a seminar where Takemusu aiki, our 'nishio branch' and a sensei teaching from a third influence each gave a one hour lesson. Quite interesting to see all the similarities and the small differences and the impact they had on each and everybodys approach to Aikido.
Anyway - Nishio sensei often said that O-sensei claimed aikido was founded in the way one moves with the japanese sword, but also that he rarely gave examples. Therefore it was up to each and everyone to do your own interpretation of this knowledge. That is one of the reasons Nishio sensei took up Iaido and Jodo, and he used this to form three distinct set of practices. One is jo-sabaki and to-sabaki - that is techniques performed very much like in tachi-waza but while holding a weapon. This is seen somewhat similar but not entirely the same in other styles as far as I know.
Also he developed a set of ken-tai-ken / ken-tai-jo katas. Paired katas where two people learn basic footwork and understanding of maai through working with weapons. Some of it has similarities to kendo-kata. I believe these are unique for aikido inspired by Nishio sensei. Other senseis have different katas along the same line, but with a different set of moves. This also underlines the changes Nishio sensei proposed in tachwaza in order to evolve irimi and technique along the lines he thought was important.
Finally he also developed a number of sword katas deviced with the purpose of giving us basic knowledge in how to move with at japanese sword. For a long time they were recognized as iaido-kata's and iaido gradings were given to his students. However this changed and Nishio sensei started giving grades in his own system. Today it is organized in a seperate association and is known as 'Nishio Aikido Toho'. It is not a prerequisit for training Aikido, but many people still train this alongside their aikido training.
The reason Aiki-toho and the ken-tai-ken / ken-tai-jo of Nishio sensei is so heavly used in Denmark and a number of other western countries or regions roots in the fact that Nishio sensei for almost two decades where the main influence here - unless you were part of the Takemusu organisation and therefore looking towards Iwama and Saito sensei instead. I think that Denmark to a certain extend was the 'sandbox' of Nishiosensei where he could see how his aikido would develop when it was not influenced very much by other styles. He constantly continued to change and adapt his aikido and related weapons systems, and I think he enjoyed leaving a new set of forms with us and then come back one year later to see how they had developed in the meantime.
I regret that the above completely neglects the fact that other styles are represented in Denmark- Ki-aikido just to name one. I just wanted to emphazise the fact that Nishio had a group of followers here that were a good testing ground for his weapons work.
There is a lot more to this story - but I hope it shed at least a little bit of light on the weapons systems of Nishio sensei.