If he became a citizen here there may be a naturalisation record for him.
The National Archives holds records for anyone naturalised up to 1986
There's some guidance on this here:
https://www.nationalarchives.gov.uk/help-with-your-research/research-guides/naturalisation-british-citizenship/
There are a few issues I think you should consider. A lot of records remain restricted until 100 years after the birth date of the individual so you wouldn't necessarily be able to access them anyway.
If he was naturalised, you might not be able to apply for citizenship of that country as he may have lost the citizenship of that country. So even if you otherwise might have been able to apply, gaining citizenship elsewhere might void that. You'd have to look closely at the laws of the individual country.
Given he was born in the 1930s it likely makes it difficult to follow up via this route anyway but may be a possibility in the future. And I suspect its unlikely to give you much more information than you already have though.
A lot of country did not have a formal registration of birth until very recently. From what you say and the fact the period he was born was one which was unstable and dangerous I think it highly unlikely there will be a record in the country of his birth either. I do think you will find this a fairly dead end.
The final option is the DNA route. But the databases for people outside the English speaking west are much more limited. Even the regional ancestry profiles for certain parts of the world are still limited. From what I've seen its still really in its infancy for the Middle East with areas huge and I suspect not really able to tell you a huge amount more than you already know. At this stage, but this may be an option in the future.
So if I'm honest, I think you have a limited number of options.
What might be out there is perhaps a religious record relating to his parents from the country of his origin. This would depend on you knowing their names. (This might be where a naturalisation record might help). It would depend on whether they had common names and whether they moved from where they were born. But its a possibility. (If you knew your great grandparents names it might help in this too). You'd need an experienced local researcher for that as its unlikely to be something you could do yourself. Or you might be able to find similar for older siblings of your father if you can't find your grandparents.
I'd say, that this is possibly your best shot at present - but tracking down a reputable local researcher in itself may be difficult. It depends entirely on which countries we are talking about.