I'm well thanks, hope you are too and enjoying your training.
Actually none of them is our branch, the closest is Hokiyama which was the continuation of Oe Senseis dojo. The current soke is Muranaga Hidekami Sensei.
Hokiyama Namio Sensei became 18th Soke after Oe Sensei's death, though there was some debate initially as Oe Sensei didn't officially appoint a successor and there were two other men who may have had claim to the title (Yamauchi Toyotake and Masaoka Kazumi) and also it seems called themselves soke. They must have stopped attending Oe Senseis former dojo after this time.
Hokiyama Sensei was Soke at the Chidokan, Oe Sensei's former dojo, from 1927, he died in 1935. After this Fukui Harumasa became 19th soke, he died in 1971.
Iwata Sensei studied Iaido at the Chidokan from August 1957 for five and a half years under the tuition of Yamamoto Takuji. Yamamoto Takuji had studied with Oe Sensei for 6 years prior to his death and Iwata Sensei notes that he wasn't so young when he studied under Oe Sensei and had learned Oe Senseis last Iaido thoroughly. While Iwata Sensei was at the chidokan he also had some instruction from the then 19th soke, Fukui Harumasa and Taoka Tsutau Sensei. Iwata Sensei was 43 years old when he began learning under Yamamoto Takuji at the Chidokan.
Iwata Sensei stopped attending the Chidokan dojo after five and a half years, he then started to receive tuition from Mori Shigeki Sensei in Matsuyama, another former student of Oe Sensei. Iwata Sensei continued his studies with Mori Sensei up until his death in 1988.
This article written by Iwata Sensei downloadable from eikoku roshukai website gives you more details about Iwata Senseis teachers and study: My Iaido
Tim and some of our other seniors may be able to give you a lineage chart for Iwata Sensei that shows his teachers and how they are linked to Oe Sensei.
Hope this answers your question.