It is pronounced with 'm' like in てんぷら (tenpura) when followed by a 'p', 'b', or 'm.'
edit: more fun facts
'n' is pronounced with 'ng' like in てんき (tenki) when followed by 'k' or 'g'
It is pronounced as a nasalized sound of the proceeding vowel when followed by 's', 'h', 'y', w', or a vowel. こんばんは (konbanwa) is pronounced kombaáwa (nasalized á).
But if you still doubt, I grabbed the first three audio recordings of "tempura" I could find in videos. All are very clearly bilabial nasals (i.e. "m"): https://www.youtube.com/watch?v=4b6glqZKVBY https://www.youtube.com/watch?v=RUslqCQd4W4 https://youtu.be/56PHtLWJh-0?t=159
Now, maybe having modeled that, when they spoke at normal speed it was an "m" sound and, not listening for it, I never heard it.
(Also: They never corrected my pronunciation of ん, which was consciously an "n" sound. They did correct my pronunciation of other things, like failing to subvocalize vowels between unvoiced consonants (I remember a long and arduous exchange in which my professor pronounced "hito" (人), and I tried to mirror it back, imagining that my problem was in not emphasizing the second syllable enough rather than failing to subvocalize the "i"). And I think that I say "input" with an "n" sound. But maybe I'm incapable of saying "input" the way I normally do in speech when I'm actually thinking about it. :P )
Maybe it's not a romanization?
In this case it is a romanization of an old Chinese-style Japanese transliteration (pre-katakana, so it reads as nonsense) of a Portuguese transliteration of a Latin word.
It's not even a round trip like anime (Latin -> ???->->->??? -> English -> Japanese -> English); but even anime got a proper romanization, it just turned out closer.
Note that this happens in Japanese when /n/ (ん) occurs before syllables starting with /b/ (shimbun 新聞), /p/ (sampo 散歩), and /m/ (sammai 三枚).
Also note that there are regional, socioeconomic, and register differences regarding whether/when this /n/ to /m/ transition occurs in a speaker's actual pronunciation.
'impossible' and 'improper' is 'in-' modified by the 'p'.
C.f. "Shimbashi", "kampai", etc. It's not an ad-hoc thing being done for "tempura".