I know that a common source for the reconstructed pronunciation of latin is Vox Latina by William Sidney Allen, but I've read somewhere that a critical analysis of latin texts and the pronunciation reconstruction started in Germany by the end of the 19th century, but I cannot find any reference on this. Does anyone know when the first works on the reconstruction of latin pronunciation began and who were the authors? Also has there been any changes/ disputes on the reconstructed pronunciation?
If you are asking simply for the earliest discussion of correct Latin pronunciation, then Quintilian (c35-c100CE) deserves a mention. INSTITUTIO ORATORIA BkI Ch4 The translation here is by HEButler (Loeb 1920). Much gratitude to Bill Thayer for the site.
On sounds lacking alphabetical forms:
Aut grammatici saltem omnes in hanc descendent rerum tenuitatem, desintne aliquae nobis necessariae litterae, non cum Graeca scribimus (tum enim ab isdem duas mutuamur), sed proprie in Latinis: ut in his "servus" et "vulgus" VIII. Aeolicum digammon desideratur, et medius est quidam u et i litterae sonus (non enim sic "optimum" dicimus ut "opimum"), et in "here" neque e plane neque i auditur;
But all teachers of literature will condescend to such minutiae: they will discuss for instance whether certain necessary letters are absent from the alphabet, not indeed when we are writing Greek words (for then we borrow two letters from them), but in the case of genuine Latin words: for example in words such as seruus and uulgus. 8 we feel the lack of the Aeolic digamma; there is also a sound intermediate between u and i, (for we do not pronounce optimum as we do opimum), while in here the sound is neither exactly e nor i.
...and on variations in pronouncing consecutive vowels:
Quia "iam" sicut "etiam" scribitur et "uos" ut "tuos". At quae ut vocales iunguntur aut unam longam faciunt (ut veteres scripserunt, qui geminatione earum velut apice utebantur), aut duas: nisi quis putat etiam ex tribus vocalibus syllabam fieri si non aliquae officio consonantium fungantur. XI. Quaeret hoc etiam, quo modo duabus demum vocalibus in se ipsas coeundi natura sit, cum consonantium nulla nisi alteram frangat: atqui littera i sibi insidit ("conicit" enim est ab illo "iacit") et u, quo modo nunc scribitur "vulgus" et "servus".
Sciat etiam Ciceroni placuisse "aiio" "Maiiam"que geminata i scribere: quod si est, etiam iungetur ut consonans.
For instance iam and etiam are both spelt with an i, uos and tuos both with u. Vowels, however, when joined as vowels, either make one long vowel (compare the obsolete method of indicating a long vowel by doubling it as the equivalent of the circumflex), or a diphthong, though some hold that even three vowels can form a single syllable; this however is only possible if one or more assume the role of consonants. [Ch.11] He will also inquire why it is that there are two vowels which may be repeated, while a consonant can only be followed and modified by a different consonant. But i can follow i (for coniicit is derived from iacit): so too does u, witness the modern spelling of seruus and uulgus.
He should also know that Cicero preferred to write aiio and Maiiam with a double i; in that case one [p69] of them is consonantalised.
I think the reference you are looking for is here: https://en.wikipedia.org/wiki/Contemporary_Latin#Pronunciation
It refers to K.L. Schneider's Elementarlehre der Lateinischen Sprache, 1819
I have not read this book and so cannot speak to specific changes in understanding, but there remain many disputed areas of reconstructed pronunciation for Classical Latin. One example is whether short and long vowels differ in quality as well as length, which is dealt with by Andrea Calabrese here.