The railway companies appear to have decided to use the "Nihombashi" spelling for the name of the stations there. So that's the spelling to use if you're searching for train times etc.

This way of writing the name reflects the fact that the "n" syllable often morphs to more of a "m" when it's immediately before a voiced plosive ("b"). **

If you want to have a consistent approach to romanizing Japanese - perhaps because you want the relationship between "Nihon" and "Nihonbashi" to remain clear - you might choose to romanize the word with an "n" rather than an "m".

I think this has been discussed in another thread, in which some people displayed an awesome level of knowledge about the alphabet. It was like super-geeky Sesame Street. Not that I'm calling anyone a grouch. You might find it if you search.

** That's a letter b in quotation marks and parentheses. It's not a attempt at a cute emoji.