How do you pronounce "heav'n"?
Often, "heav'n" occurs before a word that starts with a vowel. In that case, I think it's clearly possible to omit a syllable: there's no particular difficulty as far as I know in pronouncing "heav'n and nature" with four syllables.
(It's not totally clear to me whether the h in has might be supposed to be elided in "Heav'n has".)
In other cases, I doubt it's phonetically impossible to pronounce /heav'n/ as a monosyllable [hɛvn], but there are two issues that make this pronunciation difficult for a native speaker of modern English. One is that we aren't used to having word-final [vn] as a target pronunciation. The other is that we aren't used to hearing the difference between word-final [vn] and syllabic /vən/. It's possible to perceive vowels/syllables that aren't phonetically there. This is heavily based on a person's native language: for example, an English speaker probably won't hear any extra vowel in a word like "spit" [spɪt], but a Spanish speaker is likely to hear a (phonetically nonexistent) vowel before the /s/, and a Japanese speaker is likely to hear a vowel after the /s/. So even if you succeed at producing phonetically monosyllabic [hɛvn], it might not sound as different from /ˈhɛvən/ as you might expect.
The phonetician John Wells made a blog post about the "compressed" pronunciation of words like heav'n: "heav’nly scansion". Wells says
In modern English the word heaven, like seven, given, even, when spoken in isolation and in ordinary slow formal or colloquial style, must have two syllables. Phonemically heaven is /ˈhevən/, with the /ən/ sequence optionally being realized as a syllabic [n]. The cluster /vn/, with ordinary non-syllabic n, is not a permitted final cluster, and if there is no [ə] present then the [n] must be syllabic. (This is not the usage of eighteenth-century hymnodists, for whom “heav’n” could be, and usually was, scanned as one syllable in all contexts.)
However, like other syllabic consonants, this [n̩] is subject to possible loss of syllabicity when followed by a weak vowel. Compare seven and a half pronounced as four syllables only, ˈsevnənəˈhɑːf, or indeed heaven and earth as three, ˈhevnənˈɜːθ.
In heaven and nature sing the /ən/ of heaven is followed each time by the weak /ə/ of and. So the conditions for compression are satisfied, the word can be pronounced as a single syllable, and the notation in the score is justified.
In the recording made by Boney M, interestingly enough, we hear clearly disyllabic ˈhevən each time my score has heav’n, but compressed monosyllabic hevn for the last repetition, where my score says heaven. Mariah Carey, on the other hand, sings disyllabic ˈhevən everywhere. This may be because she’s American, and the Americans don’t seem to do nearly as much compression as the British. (Compare typical AmE ˈfedərəl federal with typical BrE ˈfedrəl.)
Walker's 1822 pronouncing dictionary says
... nothing is so vulgar and childish as to hear swivel and heaven with the e pronounced distinctly, or novel and chicken with the e suppressed.
It seems that in 1822, there were some words (chicken, aspen, patten, leaven) where the en was pronounced with a vowel—probably [ǝ]—and some words (harden, heaven, fallen, burden), where the en was pronounced as a syllabic consonant [n̩]. Today, we consider these as allophones (so they are completely interchangeable), but apparently not back then for upper-class British accents.
It is difficult to sing syllabic consonants, so presumably words like heaven were treated as one-syllable words in music, where the 'vn' was pronounced quickly between the notes (e.g., "and Heaven and nature sing" in the 1719 Christmas carol Joy to the World). All three uses of heaven in Handel's Messiah are spelled heav'n and sung on one note, for example
He that dwelleth in Heav'n shall laugh them to scorn,
however, listening to these sections on the internet, modern singers use two syllables much of the time, showing that you're not the only one who's confused about how to pronounce heav'n.
In poetry, Shakespeare treated heaven as either a one- or two-syllable word, depending on which he needed to make the lines scan. For example, we have
Sometime too hot the eye of heaven shines,
from Sonnet XVIII, and
To shun the heaven that leads men to this hell,
from Sonnet CXXIX. To make these lines scan, heaven must be pronounced with two syllables in the first and with one syllable in the second.
Shakespeare didn't do the same thing with burthen/burden, so maybe the distinction between syllabic [n̩] and [ǝn] goes back to the 16th century (although some words like burthen would have had to change their pronunciation).
So my guess is that heav'n is pronounced with a syllabic [n̩]: [hɛvn̩]. It still seems like two syllables, but it's easier to compress into the space of one syllable than [hɛvǝn]. If you know French, think of words like livre /livʁ/ and table /tabl/ where the /r/ and /l/ are kind of tacked onto the end of the syllable.
As a Brit, this is my two penn'orth. It is kind of impossible to pronounce 'heav'n' as one syllable. However, it does sound quite different to 'heaven'. As I'm saying it now, it sounds similar to 'hen', except that I'm lifting my lower lip as though to pronounce the 'v', but there's no exhalation through my mouth, so the 'v' is not actually pronounced. So, it becomes a more nasal-sounding word than 'heaven'.
I have no idea if that makes sense!