Familiarity with the possible words helps. Even a native speaker is left struggling when listening to someone with a strong accent. It's knowing what the word might be that allows them to make a best guess That is harder with proper nouns, names and place names, as they are much more random. Face to face or on video you can watch for the lips closing for m and being slightly open for n.

Native speakers have trouble spelling words from listening to them. Online sources of song lyrics, transcribed by people who simply listened to the words, are full of mistakes. For dictionary words, we rely heavily on context. For names of people and places, which not only aren't phonetic but are often borrowed from non-English sources, it's often just a guess. Over the telephone, for example when making a hotel reservation, even though "Smith" is the commonest of all surnames, I am _always_ asked to spell it out. I don't know how IELTS is graded, but, frankly, in proper names, I wouldn't worry about it. They can't POSSIBLY expect you to do well at a task that native speakers can't do well. "M" and "N" are similar in sound. Ventriloquists (performers who create the illusion that a doll is talking) don't want to use the "M" sound, because it makes their lips move in an obvious way. So ventriloquists simply substitute N's for M's throughout. They will say "Ny nane is Nary" and listeners actually think they are hearing "My name is Mary." In informal U.S. written English, you can write "(sp?)" as an abbreviation meaning "I'm not sure of the spelling." I don't know whether or not you should do that on IELTS. My instincts would be to go ahead and do it,

My take on your question is along the same lines as Peter's. I'm not a linguist but I would add that different languages have different vowel and consonant stressing patterns. This contributes to the rhythm of the language. You'll get it once you begin to listen to native speakers and imitate them to the best or your ability.

I think once you practice more with verbal speech you can tell the difference. The "n" sound puts your tongue on the roof of your mouth if said properly. The "m" sound is your lips being in contact and not your tongue--so they do make different sounds. Say the food name "corn" with a M. "Corm". It sounds different. Now say "comma" with n, as "conna"--two different sounds. Now say "mom" with n, as "mon"--sounds different again if said properly.