Perhaps someone from southern Africa can answer this question better than I can, at least as far as South African English is involved.
Let’s start with the voiceless clicks, the ones spelt in Xhosa, Zulu, and other Nguni languages as c, q, and x. They are dental, (post)alveolar/retroflex, and alveolar lateral respectively. In current IPA they are written ǀ ǃ ǁ; in earlier IPA they were ʇ ʗ ʖ. My own impression is that the usual anglicization of these sounds is k. (Recall that clicks involve a simultaneous velar closure. A voiceless click is articulated during the hold phase of a voiceless velar plosive, k. Some people like to show this explicitly in the phonetic notation, thus k͡ǀ etc.)
Thus the name of the Xhosa people and language is pronounced in English with a simple k in initial position, ˈkɔːsə or ˈkəʊsə. In the Xhosa language itself the initial consonant is ǁʰ (old ʖʰ), an aspirated alveolar lateral click. This name seems to be the only anglicized word with an original click that is reasonably familiar outside South Africa.
Some historians will know about an event referred to as the Mfecane (also Difaqane, Lifaqane), the “scattering” of non-Zulus fleeing the Zulus in the early nineteenth century. In Zulu that’s mfɛˈ|aːnɛ (I don’t know what the tones are). I have heard it called əmfeˈkɑːneɪ, though I’m not sure whether that is the usual anglicization — the word is not even mentioned in Branford’s Dictionary of South African English (OUP 1980).
Anthropologists and linguists may know about the !Kung (or !Xũ) people and language. They live in the Kalahari/Kgalagadi (pictured above). As Wikipedia tells us,
To pronounce "ǃKung" one must make a click sound before the 'k' sound, often represented in texts as an exclamation mark.Strictly that should be a (post)alveolar click, old IPA ʗ, with affrication of the pulmonic-air velar component. But I think people (or non-phoneticians at any rate) usually give up and just say kʊŋ.
What about voiced clicks? These are the ones spelt in Xhosa and Zulu as gc, gq, gx. They are articulated during the hold phase of a g. I can’t think of any words or names with a voiced click that are used in English.
Then there are the nasal clicks, made simultaneously with a ŋ. These are the ones spelt in Xhosa/Zulu as nc, nq, nx or (breathy-voiced) ngc, ngq, ngx. The only relevant example I can think of is the Ndebele surname Ncube, which British newsreaders pronounce as ˈnuːbeɪ, ignoring the click. The OBGP calls this the ‘established anglicization’ of this name, and comments as follows.
I have never heard anyone speak of the lawyer Bulelani Ngcuka, or of his politician wife Phumzile Mlambo-Ngcuka. Perhaps some South African reader has? (That’s Xhosa ˈŋǀʱuːkʼa, old notation ˈŋʇʱuːkʼa, tones not known.) He was a member of the law firm founded by Griffiths Mxenge (presumably mˈǁɛŋɛ, old notation mˈʖɛŋɛ), who was assassinated by the apartheid police in 1981. I haven’t heard that name pronounced in English either.
So, summing up, the best tentative rule I can offer Jongseong is: to anglicize a click, ignore the velaric (click) component, and pronounce the rest, which is necessarily some kind of velar pulmonic consonant. Voiceless clicks become k, voiced ones ɡ, and nasal ones ŋ. But in initial position (which is where clicks are usually found), the last-mentioned obviously becomes n.