I visited Nakhichevan in March 2018 and there was a weekly service running between Tabriz (Iran) and Nakhichevan (Azerbaijan) using Iranian stock. There is very little information available and i wasn't even aware of whether or not it was running until i got there.
There is also a daily domestic service as well within Nakhichevan using Azerbaijani stock. I took a photo of the timetable board showing both the domestic and international train times to/from/within Nakhichevan which i will try and find.
This was in March 2018 so it could have changed now.
I have found this much more recent article here -
https://en.trend.az/business/tourism/2881250.html - which was posted just after i visited which says there is a direct train continuing beyond Tabriz to Tehran and then Mashhad after that. When i visited it only went to Tabriz and terminated there.
Nakhichevan is a very interesting place to visit. It's quite different from the main part of Azerbaijan and they certainly aren't used to seeing tourists there.
The closure of the Armenia and Azerbaijan border certainly makes travelling in the region very difficult. There are so many closed railway lines because of this. Also you have the disputes in Artsakh and Abkhazia and South Ossetia which further complicates things.
Hopefully railway services around the Caucasus and Central Asia regions will gradually improve in the future.