Would it not be appropriate to translate this as, "No thanks, I don't want any"? It sounds more natural to me in English to say "No thanks" with a negative rather than "Thanks".
"No, thanks." is perfect - that's what we say to avoid an offer, just like "Köszönöm, nem kérek."
If it said nem kösönöm then you could write no thanks. You could say that if you were speaking but it is not the translation of the given sentence.
In Tatar langauge the word "kiräk/кирәк" (in Bashkir: käräk/кәрәк") also means "want" or "need". For example, the phrase "Miŋa maşina kiräk/Миңа машина кирәк" means "I need/want a car".
Interesting. I noticed that some of the few words I recognize in Hungarian are ones I know from Russian, which must have also gotten them from a Turkic language. For example, "Wednesday", "szerda", "среда", which at least appears vaguely similar in Turkish, "Çarşamba".
среда means the middle of the week in Russian, like Mittwoch in German. It does not mean anything in Hungarian, except the name of the 3rd day. You may recognize the old Persian word 'bazaar' in the name of the 7th day: 'vasárnap' in Hungarian, i.e. 'market day'. This came via Turkish, in which Sunday is still called 'Pazar'.
I don't think that the word "szerda" is Turkic origin. It's pure Slavism. "Çarşamba" is originally a Persian word, as well as the other days of the week in the most of the Turkic languages. Magyar nyelv has a lot of Turkic words, i think mostly from Bulgaric branch. But there are also words that must understand every Turk. For example (Magyar-Kazakh-English): Alma-alma-apple, anya-ana-mother, balta-balta-ax, béka-baqa-frog, homok-qum-sand, tenger(bulgaric)-teñiz-sea, kék-kök-blue, sárga-sarı-yellow, ökör & bika -ögiz & buqa-ox & bull, teve-tüye-camel, szel-jel-wind, szakáll-saqal-beard, macska - mısıq - cat etc. :)
"Thank you, I would not like any" seems like a perfectly good translation to me, especially if 'akarok' is 'I want'. Unless I'm mistaken...
Where do we get the any from? Is that just the assumed translation into English as it would be different to say I do not want?
I don't understand the "nem kérek" part. We had a sentence earlier with akarok meaning I want. I'm guessing nem kérek isn't a literal translation of I don't want any.