Hello,
It is my understanding that upon purchasing a property, the transfer tax calculatation is done by the NAV after the county Land Registry sends them the documentation confirming the new ownership. Within a few weeks, they should mail a letter to the purchaser confirming the amount due and there is a ~15 day period to pay it.

In my case, it's now been about 10 months since I received ownership confirmation from the Land Registry and I haven't heard anything from the tax authorities. I don't reside in Hungary so I do not have access to the online citizen portal to check anything there (I'm in another EU state), but I have not received anything in the mail to either the foreign or Hungarian address.

I don't want to run into problems down the line with fines or worse, has anyone ever successfully contacted the NAV in English? Or perhaps faced similar delays?

If you go to their website nav.gov.hu, you can select English as an alternate language and it say on the main page they want to help non Hungarian speakers with the rules and processes. Perhaps if you try contacting them in English they will be happy to sort out what has happened.