A lot of older hymns and songs in English appear to have rhymes that, to modern ears, don't actually exist except on the page – 'blood' and 'good', 'love' and 'move', for example. Did these rhyme at the time but have diverged, or were they always a bit contrived?

I'm writing this as a historian interested in late medieval England, so my knowledge of the topic of historical phonology and linguistics of English is limited to its use in my subject area.

For a more in-depth linguistics focused answer you might try posting your question to r/linguistics.

However, for a brief answer, yes, rhymes that don't quite work in modern English would have sounded differently to the ears of those reading them contemporaneously to their creation. Linguistic historians use these rhymes as supporting evidence that a change in pronunciation has occurred over time.

As you touched on in your post, a key part of figuring this out is knowing the context of the form and structure of the verse. If we know the pattern for which the author typically rhymes, then we can make pretty good guesses about what other words should have rhymed as well.

For example, when looking at Chaucer's Canterbury Tales, we see there is a rhyming couplet structure of

aa, bb, cc

Whan that Aprille with his shoures soote
The droghte of Marche hath perced to the roote,
And bathed every veyne in swich licour,
Of which vertu engendred is the flour;
Whan Zephirus eek with his swete breeth
Inspired hath in every holt and heeth

If you listen to the [audio recording] (http://facweb.furman.edu/~mmenzer/gvs/chaucer.htm) of the passage, you get a sense of how the rhyming provides clues to pronunciation. Again, context is key, because some medieval poets like Chaucer tended to be faithful to be strict in sticking to rhyming conventions, where others, such as many romances, varied more often.

Chaucer was writing in Middle English during the late 14th century. Beginning in the late 14th century and lasting until the 18th century, English underwent what is known as "the Great Vowel Shift." Peaking in the 1400s-1500s, the result was massive changes in how words were pronounced, particularly a shift away from the long vowels of Middle English. Depending on where a text falls within this period, you might expect pronunciations to be more or less familiar to a modern ear.

There is quite a bit of debate as to the causes of this shift, including the increased migration and urbanization of the population due to the impact of plague, and the introduction of printing.

You may have heard of David Crystal's work on the Original Pronunciation of Shakespeare's work, but if you haven't I'd recommend checking it out. There are several videos on Youtube that offer a quick introduction to this kind of work.

Shakespeare, rhymes, and Original Pronunciation

Shakespeare: Original pronunciation (The Open University)

Donka Minkova, A Historical Phonology of English

Ben Freeman, From Old English to Standard English: A Course Book in Language Variation Across Time