I have some faith in Snopes.com as a reasonably impartial fact-checker. Here's what they say about the Azov movement:
https://www.snopes.com/news/2022/03/02/what-is-ukraines-azov-battalion/
tl;dr: numerically small but disproportionately prominent in news coverage (a bit like UKIP were in the UK).
The roots of the organisation are definitely far-right and some are openly neo-Nazi. There was a Newsnight investigation of these people, and associated groups, a few years ago.
It certainly strikes me as weird (to put it mildly) that the Azov Battalion is an official part of the Ukraine national army - a bit like having a KKK division in the US army, or an EDL regiment in the UK.
In summary (my opinion): the extent of neo-Nazism is being overstated, but it is not nothing, is probably not negligible, and is quite likely to increase as a direct consequence of this conflict.