The question of how to pronounce anusvāra in syllables like oṃ and hūṃ is a question I get asked quire frequently. There are two answers to this, and I'm not sure it's any longer possible to decide which is right.
Anusvāra means 'after-sound'. It is was originally written, in the Brahmī scrpit as a small dot on the right-hand side of a syllable, but soon migrated to the top as in modern day Devanāgarī: e.g. ta त vs taṃ तं In Roman script we use an m with an underdot ṃ; or some times an overdot ṁ; or an n with a tail in older books: ŋ. There is also the anunāsika 'from the nose' which is a more general word for nasal sounds including consonants like ṅa, ña, ṇa, na and ma, and the nasalised semivowels e.g. lṃ. The anunāsika is indicated with candra-bindu in Devanāgarī: la ल vs lṃ लँ. Note that in modern languages such as Hindi the anusvāra and anunāsika are used interchangeably, but the anunāsika is used with seed-syllables where is adds to the symbolism.
What the anusvāra does when added to a syllable is nasalise the vowel of the syllable. Even though the sound is indicated by an additional letter in Roman script (the modified m) it is only indicating a change in the quality of the vowel sound. When speaking the air coming out of your lungs to make the sounds usually escapes through he mouth where the tongue and mouth shape the sound into distinct units. With a nasal sound, however, some or all of the air escape through the nasal cavity - which changes the timbre of the sound to produce the distinct nasal drone or buzz. You can experiment with the different sounds by pronouncing pairs of sounds that are only different because on is nasal: pa and ma have more or less the same articulation except the ma is nasal. Similarly ka and ṅa, ca and ña, ṭa and ṇa, or ta and na.
Now a vowel is produced with an open vocal tract - one doesn't interrupt the flow of air with the velum, tongue, or lips. Nasalising a vowel sound only requires a lowering of the velum (the soft palate at the rear of the mouth). The sound is used in English as the end of words like sung, sing, sang.
However there is another way to make a nasalised sound which is to bring the lips together at the end of the syllable, forcing all of the out going air through the nose to make a nasal humming sound. This is different from the ma syllable which begins with closed lips that open suddenly to release the built up pressure With the labial anusvāra the lips stay pressed together.
Both of these methods are common, and as far as I know there is no way to decide which is more correct.
So oṃ can be pronounced two different ways: ong and om (with no release at the end). With oṃ the latter is more common, but when one is learning Indic languages one hears a range of pronunciation of nasalised vowels.
Similarly hūṃ is pronounced as both hoong and hoom. The ū sound is a bit like the English vowel sound in 'hoot', but more open (i.e. the lips are not so bunched up). Make the face for hoot and then relax the lips. But please note that accents are all about how you pronounce vowels. You tend not to hear your own accent, so you pronounce vowels quite automatically. So describing how a vowel sound should be pronounced is usually folly at best. For ū the International Phonetic Alphabet sign is /uː/.
So we see oṃ pronounced more usually with the -m sound, and seed syallbles such as tāṃ, or hūṃ with the -ng sound. Take your pick. What most people do is follow the people around them. And then at least we're all doing the same thing, and what is language but a social convention anyway?
The idea of reproducing the "original" pronunciation is probably just a fantasy. Sometimes we can reconstruct pronunciation from rhyming patterns in verse, but the fact is the we just don't know and we never will, because we can't hear the ancient speakers. Pronunciation changes. Modern day Hindi speakers do not pronounce the vowels exactly as Sanskrit would seem to demand either. For instance they flatten the ai vowel so that it sounds like e. So the "original" pronunciation is lost. We must be content to do our best, and produce sounds that seem to fit what is written.
As I showed in my article on the 100 syllable Vajrasattva mantra in the Western Buddhist Review the errors introduced into mantra recitation are sometimes due to mis-reading rather than mis-hearing. So we are not the first to struggle with these issues!