This might be a good one to field at /r/linguistics, but I’ll give it a shot. I can only really speak with any authority on Middle English dialects of roughly the twelfth through fifteenth centuries, but some of the linguistic principles and methods are transferable and might be useful when thinking about other languages and periods.

There is indeed a field of study dedicated to the reconstruction of variations (both geographical and temporal) in the spoken and written languages of the past: historical dialectology. In Middle English, the main tool for our understanding of dialect is scribal variation. Generally speaking, Middle English was not standardised; when scribes copied texts they usually transposed the spellings (orthography) in line with their own usage, which broadly speaking will have reflected their pronunciation (phonology). This gives us a resource through which to try to understand historical dialects. Two hugely influential projects, A Linguistic Atlas of Early Middle English and A Linguistic Atlas of Late Middle English sought to do something with this material. By collecting manuscripts that can be localised and dated based on internal evidence, these studies assembled and plotted a huge amount of data that can be used to make generalisations about dialectal usage.

As a totally arbitrary example, here are three forms of Middle English many as represented in LALME (c.1350–1450). We can fairly clearly see the ubiquity of the form many, but also a distinctly southern usage of meny and a broadly north-western usage of mony. The collection of thousands of similar data points from hundreds of manuscripts allows historical linguists to get some grasp of local variation in orthography and use that understanding as the basis for phonological study. It is possible to take manuscripts with atypical spellings that can’t be placed on the basis of internal evidence alone and fit their usage against these dot maps for a reasonably reliable idea of their point of origin.

I should note that LALME and LAEME are not the be-all and end-all of Middle English dialectology, because the available evidence comes with a number of inherent stumbling blocks. Both studies necessarily collate data on manuscripts produced a century apart – and it is sadly true that dating manuscripts on palaeographical grounds is itself a less than exact science. The degree of detail to which orthographical variation can represent phonological practice is open to debate. It is demonstrably true that not all scribes totally ignore the spelling of their exemplars in favour of their own personal usage, which potentially skews the data. Most significantly of all, of course: scribes and books travel. Even if we know that a book’s first owner lived in, for instance, Cheshire, we cannot say with any certainty that the book was actually copied there, or that the scribe lived there or had grown up there, or that their exemplar was locally sourced. None of this invalidates the studies, but must encourage us to carefully handle the data and look for other kinds of confirmation before making any bold claims about the nature of Middle English dialect.

Fortunately, there are other kinds of confirmation which historical dialectologists can turn to. Although Middle English as a whole was not a standardised language, some influential studies have argued convincingly for the existence of some consistent local standards (perhaps more consistently taught and practised than geographical dialectal variation) that have allowed scholars to build a more developed picture of particular dialects. Perhaps most famously, M. L. Samuels (‘Some Applications of Middle English Dialectology’, English Studies 44 (1963), 81–94) identified his so-called “Four Types”, one Central Midlands dialects and three London dialects, that present consistent orthographical variants across substantial manuscript corpora. The other famous example, the “AB Language” was first noted by J. R. R. Tolkien (‘Ancrene Wisse and Hali Meiðhad’, Essays and Studies by Members of the English Association 14 (1929), 104–26) in some anchoritic manuscripts. These and similar studies allow a more focused look at particular usage of language at discrete points in place and time.

We are also occasionally afforded a sense of historical dialect by sheer good fortune. The Ormulum, a book of verse homilies copied in the mid-twelfth century by a monk called Orm, is of perennial interest to scholars of early Middle English. For reasons probably best known to himself, Orm decided to re-imagine the whole system of English spelling. He meticulously uses doubled consonants to indicate vowel length, distinguishes between Ȝ and g, and puts in accent marks to eliminate ambiguity. Orm’s is a fascinatingly ambitious and as (far as we can tell) single-handed attempt to solve the problem that English contains more phonemes than the Roman alphabet has letters. This deliberate attempt to represent phonology through orthography, combined with the even stresses provided by his use of Classical metre, makes Orm an invaluable snapshot into the language of the twelfth-century East Midlands.

There are various other miscellaneous artefacts like this that linguists can commandeer as evidence. Chaucer famously, in The Reeve’s Tale, portrays some Northern students as using variant words and pronunciations that accord with LALME’s findings in Northern manuscripts – and scribes, for the large part, retained these variant spellings. While I’m on the subject of poetry, rhyme can of course be used to tell which words sound alike, which is occasionally useful in clarifying ambiguity.

It is also possible to “work backwards” toward historical phonology, although to be honest I lack the linguistic expertise to talk about this in much depth. The Great Vowel Shift that took place in the fifteenth and sixteenth centuries (which probably accounts for the most significant difference between Middle and Modern English) is recorded across centuries of orthographical variation in personal letters and other documents. Far better linguists than I have studied how these spelling changes represented ongoing changes in vowel pronunciation and by “retracing the steps”, as it were, have been able to approximate pre-GVS phonology.

What I suppose I should stress is that there is a lot of disparate evidence for Middle English dialectology, almost all of it flawed in some regard – most pressingly by the intangible barrier between spelling and speech. But by taking all this evidence together, linguists strive toward the most complete picture of the languages of the past that can be attained.

If you’re interested in knowing about this in more detail, or are looking for things like lists and tables of vowel sounds with comparisons to Modern English pronunciation, I strongly recommend Jeremy J. Smith and Simon Horobin, An Introduction to Middle English (2002). It is a very short, accessible and well-illustrated overview of what Middle English looked and sounded like, and how we know it.