As a bit of a follow up, would it be considered completely inaccurate to cite The Apology of Socrates as a trial transcript?

I'm not going to claim this is the earliest, but I had to read the records of the trial of Domenico Scandella in graduate school. He is probably best known as Menocchio, and his story is most famously (and mostly incorrectly, I would argue) told in Carlo Ginzburg's work The Cheese and the Worms.

They're not a transcript really, but rather the notes of the inquisitors, as I recall, although Scandella's words are recorded in a form. It's been a minute since I read them, so I'm not going to offer further details, since this isn't even remotely close to my specialty. I'm sure someone else here has read them more recently than myself.

It depends on how tightly you want to stick to your definition of 'transcript'. If you're talking about the literal record of what was said at trial, you're probably talking about the later middle ages. The Trial of Joan of Arc in 1431 (translated W.P. Barrett, The Trial of Joan of Arc, (New York, 1932)) is the first real trial I can think of, although we have transcripts of inquisitions of heretics performed by the Church going back to the early 1300s (check out Emmanuel Le Roy Ladurie's Montaillou, it's the easiest way to access them.) These weren't however, strictly speaking, trials.

If you allow a bit of a broader interpretation of the term transcript, we have court cases (summaries and verdicts) preserved from throughout the Middle Ages and several famous speeches which were delivered in court cases in antiquity have survived. Politicians in Ancient Greece and Rome often earned their reputations as orators in law courts and their speeches were often revised and published subsequently. Demosthenes (Athens) and Cicero (Rome) are probably the most famous orators whose court speeches have survived, although of course they provided only a small glimpse into the whole trial process. In fact it's important to note that in all the cases mentioned, the words spoken were translated, revised and mediated by the scribes and the transcription process as a whole, so they must all be approached with caution!

I highly recommend The Tyrannicide Brief by Geoffrey Robertson. It talks about the trial and execution of King Charles I, the legal reforms that were put in place during the Interregnum, and the trials of those who put the king on trial when the monarchy was restored. It's written by an attorney, so it delves deeply into legal processes. One of my favorite moments is when one of the judges steps down from the bench, testifies as a witness against the defendant, then gets up on the bench again.

I would have thought the various Sagas of Icelanders would also meet you criteria. Not as old as Socrates, granted, but the mediaeval Icelanders had a sophisticated legal system. I assume there are other examples, but the only one that springs to mind is Njáls Saga, which includes a depiction of an Icelandic court case circa 1000, though the text itself dates from the 13th century.

It depends on where you're looking. In southern Europe, where the Roman heritage was strong, notaries were often hired out to copy down court proceedings. Such archival series in Italy go back to the mid-to-early 13th century; in southern France (notably Marseilles) they go back to the mid-13th century. These are pretty darn close to transcripts: they were produced for litigants (written instruments and depositions were very important in Roman procedural law; witnesses were questioned privately, but after that copies of what they said had to be made available to both parties) and faithfully render depositions, albeit translated into Latin from (usually) a vernacular. Ecclesiastical courts also drew on Roman procedure (the fusion of Roman-inspired procedure and canon law is often called "Romano-canonical procedure"; back in the day it was called the ordo iudiciarius), so for certain jurisdictions we have very dense records that go quite far back--for example, court papers (depositions included) covering most of the 13th century survive from the Arches Court (province of Canterbury). But the notaries who copied down depositions certainly simplified what was said, imposing formulas. To a certain extent, however, witnesses may have given formulaic testimony. The general rule is that the earlier one goes, the more detailed and less formulaic testimony becomes, and vice versa, which may reflect increased professionalization or witnesses who were increasingly savvy when it came to giving testimony--probably both. The same goes for canonizations, which used Romano-canonical procedure to, effectively, put the reputed sanctity of an individual on trial; reams and reams of witness testimony were gathered and compiled into massive dossiers submitted to Rome, many of which survive. These go back to the early 13th century, when the papacy first started taking over the canonization process (and around the same time, it started enforcing the ordo iudiciarius in ecclesiastical courts).

Before that, we mostly have records of judicial decisions and arbitrations, which often contain a description of the case in varying degrees of detail. This goes all the way back to the Carolingian placita. (For more on this, see Davies and Fouracre, The Settlement of Disputes in Early Medieval Europe.) Many cases were preserved in monastic archives, and so these tend to relate to cases that ended in the monastery's favor, concerning land that the monastery held. These kinds of records, while not really "transcripts," go very far back--even to Roman Egypt. (See Kelly, Petitions, Litigation, and Social Control in Roman Egypt--though this deals with cases that survive from late Roman municipal archives, the gesta municipalia.)

There is a very large amount of primary English legal history available thanks to the efforts of the Selden Society, which publishes various annual volumes of medieval court documents - mainly plea rolls and year books. For example, I have a volume from 1973 in front of me entitled "The Roll and Writ File of the Berkshire Eyre of 1248." It contains the plea rolls, some of which are fairly detailed but not "transcripts," both in the original Latin and translated to modern English. The entries in the rolls commonly contain the identities of the parties, the legal issue, the arguments of the parties, and the judgment. This information is roughly comparable to the detail of information from the Old Bailey website. There are hundreds of these entries in this volume alone.

Another example is the Selden Society volume from 1969, "Eyre of London, 14 EDW. II, 1321, Vol II," which is part of a series of volumes on the year books from Edward II. This volume contains significantly more information about each case, including reasoning by the various judges. That's because yearbooks are an early form of law report.

Although I don't have them handy, some Selden publications go as far back as 1066 or earlier. Some of these publications are out of copyright and available on google books, for example "Select Pleas of the Crown: AD 1200-1225 Vol I". For more recent volumes, any decent academic law library should have a selection if not a complete collection. Some primary caselaw sources are also available online from the Ames Foundation: http://amesfoundation.law.harvard.edu/

A little bit of history about the practice of recording litigation information in English medieval history: Technically, all that needs to be preserved is the outcome of the case - party A is liable to B for X, judgment is such and such injunction or fine. For the most part, that's what plea rolls are, though some do have significantly more detail, especially for important cases. They were written by clerks and kept by the judges until a centralization effort was made in the mid 1200s. More detailed information is available from about the mid 1200s or so because unofficial reporters - observers of the court, typically apprentice lawyers - would record information about the facts of the case, arguments of the parties, and opinions of the judges. These reports appear to be generated for the purpose of learning (or demonstrating learning) about law and procedure, and were useful in assisting memory of precedent cases. Many of these reports are rather readable, as they contain commentary about the process, the actions and tempers of the judges, jokes, etc.

My understanding is that significantly more detailed information on court proceedings is available in continental Europe, especially with regards to the church, which probably did keep actual verbatim transcripts for some cases, especially regarding heresy - but others would have more knowledge about that.