I've read quite a bit about the Arian Controversy over the years, though a lot of it open source and quite dated. There's a lot written about the theologic dispute surrounding Arianism, Semi-Arianism, Wulfila, etc., but I haven't read much about the organisation of Arian Christianity. Do we have any sources on if and how the organisation of Arian Christianity, especially later Gothic and Lombard Arianism, differed from that of Nicene Christianity?
My specialty is fourth century trinity debates, and so Arianism in the context of the Roman Empire. On that score, let me say that there is no separate or distinct 'Arian' church within the empire. Organisationally you would just say that some churches or sees were Arian in their theology (though we tend not to use the label 'Arian' in recent historical scholarship because it's a mislabelling for most of the positions through the period.
I'm not well acquainted with 'Arian' Christianity in its Gothic and other 'barbarian' forms, and will leave comments on that to someone more qualified.
Edit: I should say that the quite late in the century Heteroousians aka Anomoioans did end up forming a parallel schismatic church structure of their own. Like many such groups labelled heretics, their church structure and organisation attempted to mirror the majority church , though in a smaller scale.
We do have some idea of how the Vandal church was structured in North Africa during the fifth century, mostly drawn from Victor of Vita's "History of the Vandal Persecution". Here's a good summary:
[The Vandal Church] had its own patriarch of Carthage, and bishops, though probably only a few. For as far as we can see there was no attempt to set up a bishop in every city, and the bishops we hear about usually do not have the name of a city attached to their title. The Arian bishops supervised parishes. These presumably covered areas where Vandals were settled. The services were at least partly in Vandal “German”, otherwise Huneric would not have demanded the right to use their own language in services for the Arians in the East…unlike the civic bishops of the Empire, and the orthodox bishops in even some of the successor kingdoms, the Arian bishops in Vandal Africa were not given any secular responsibilities.
The Vandals confiscated some churches together with their property for use by their own Arian clergy….The Vandal authorities were strongly concerned that there should be no Catholic proselytizing among the Vandals themselves. Members of the Vandal gens who were seen entering a Catholic church — note that such people existed — were attacked, and if they were women, publicly shamed.
-- Liebeschuetz, J.H.W.G. 2003, "Gens into Regnum: The Vandals" in Regna and Gentes: The Relationship Between Late Antique and Early Medieval Peoples and Kingdoms in the Transformation of the Roman World, 55-85.