It strikes me that this remains true, at least for those involved in law, medicine, or any other highly technical field. The amount of knowledge that lawyers, doctors, judges, etc. memorize is astounding.
And that’s not even counting the Tolkien trivia!
Wax tablets were used in 12th century France, as well. I’m not sure about their use through the rest of medieval and Renaissance Europe, though. Apprentices to artists learned to draw on was tablets in Renaissance Italy, so they were used by some people at least until the fifteenth century.
It’s not that I’m shameless that way – it’s just that I’m confident that I also talk enough rot so as not to be in any danger. 
For the main English courts, the short, simplistic answer would be 1907. But that would be to gloss over an incredibly complicated story dating back several centuries. And that story only makes any sort of sense if you first realise that the main issue for the courts was usually not whether they recorded their own proceedings but rather which records they recognised as being valid for citation as precedents.
From the seventeenth century journalistic accounts of major trials begin to appear in England with what are generally accepted as being reasonably accurate transcripts of the proceedings. What had changed was that by then they did have forms of shorthand that were more or less up to the task. This meant that by the eighteenth century various commercial publishers were printing series of unofficial trial reports. But those were not necessarily complete transcripts. What got included would depend the particular court, the type of case, commercial pressures and the vagaries of the individual publication. Longer was not obviously better - it was more expensive and what you might think would be the most interesting bits, such as the cross-examination of witnesses, were what the lawyers who bought them were least interested. In time, some courts began to recognise particular reports as having semi-official status, even although they were still private publications, but there was no simple pattern involved, with much depending on conflicting interests among and between reporters, lawyers and judges. However, by the late nineteenth century the system was becoming more formally organised.
It might seem odd that the courts felt no need to compile their own official transcripts, but, for the very reason that there was no tradition of them doing so, they had never had any difficulty in accepting that certain types of unofficial records could be cited as evidence for what had been said by judges or lawyers in earlier cases. The classic type of such records were in fact those notes made by respected lawyers (including some from the sixteenth century) recording those points they thought most interesting about the proceedings in particular cases. The issue with the shorthand reporters was therefore always not that they were unofficial but about whether they were as authorative.
One other point of note is that in those days lawyers tended to have a lot less contact with their clients or deep factual investigation than we do now – IIRC, the great barristers would get a few minutes of factual recitation on their way into Westminister, and they’d extemporize from there. What I write for a single meeting with a client might cover more information than a barrister got for a dozen of his clients’ entire cases.
–Cliffy