Medical School in Hungary

One of my kids was thinking of medicine and I looked into it some years ago.

I couldn't answer for sure about grants or scholarships but my gut feeling is that there's nothing you could apply for other than loans.    However, if you are an "EU" student by virtue of your birthright, you might be able to even get lower fees in other places, like Scotland (plenty of excellent universites) or Germany (they also do medicine in English I guess).  They would not charge you as a foreign  national if you had an EU passport.  You can also apply for student loans in the UK as an EU student (all EU nationals are treated equally, I don't see why they need to know you are dual national).

Hungary has a good reputation and history for medical training.  A lot of countries and regions send students here to study - Norway, Africa, China, Middle East and even Iran.  African countries used to send medical students to Moscow to study but since there was a series of racially based attacks on African people in Russia, there was a move to use other ex-communist countries like Romania and Hungary.  There's something of a legacy here from the communist past that build up "relations in socialism" by having overseas students from certain countries.

One area which might be an issue is if you return to the USA to practise and do not have a recognised degree which would allow you to obtain a license without further training or even worse, re-training. I understood this was necessary to obtain malpractice insurance rather than anything academic. My interpreter's husband was an oncologist and had to do a supervised training residency at a US hospital before he could practice there. That's even though he was a 10+ years experienced mid-career professional anyway.

I read some Caribbean countries have medical programmes which are recognised in the USA.  Maybe such things also exist here too but I haven't looked into it further as that part wasn't on my radar.  All EU doctors can work in any other EU country (UK now requires a language test).

Both those universities - Semmelweis and Debrecen - are top class medical schools.  Wherever you go, you'll need to learn the language (they give you courses) as you need to be able to talk to patients.

Since timing may be important in your case, you should ask the consulate how long the process will probably take, from submitting documents in proper form to receiving your Hungarian passport. Remember that everything gets sent to Budapest for processing.