Many years ago, I won an Open Scholarship to St John's College Oxford without the support of my school and without being taught for the Scholarship examination (and I still have the letter from the school to prove it). I took the Scholarship Examination in the fourth term of the sixth form, against competition some of whom had taken their A levels (and in many cases also done S levels) and were in the seventh term. I chose St John's specifically because it was the top college academically (on the basis of average performances in the Norrington table over the previous two years, which was the only information about academic quality which I had at the time). I had an article published about this in the Times Higher Education Supplement ("Exams as escape routes", p.14, 25th March 1994) which was up on the noticeboard of the Cambridge Department of Pure Mathematics and Mathematical Statistics (DPMMS) for several weeks and was discussed in the common room there. I am aware of other cases where people have got Scholarships to Oxbridge without formal preparation for the Examination, but none which satisfy all three conditions of being a fourth-term applicant, not being taught for the Examination,and being told not to apply by the school. Something which may not be obvious is that was a difference in the situation of a candidate between the cases where a school can't teach and where it won't teach. In the first situation,the applicant may have access to solutions to past questions written by students who applied in previous years, or he may be able to collaborate with a colleague in the same class. If the school wishes to support the application but for some reason cannot offer any formal preparation, then it can ensure that the A level Mathematics and Further Mathematics syllabi have been completed by the time of the fourth term examination. I have heard of all of these happening in practice. I have also heard of two cases where someone whose school refused to support his application had private tutoring. The situation today for someone whose school can't or won't teach him for the Oxford Aptitude tests or the Cambridge STEP is very different to mine : there are now many good problem books, free resources on the Internet, Internet discussion forums, and free group preparation courses available from Oxford and Cambridge. My understanding from what I have read in the national press is that the Oxford Entrance Examination was abolished in 1995 because it was felt to be unfair to those pupils who were not prepared for it (see, for example, https://www.theguardian.com/education/2000/may/24/highereducation.oxbridgeandelitism).
Many years ago Hertford College, Oxford introduced a scheme under which some applicants could be given an unconditional offer without taking the Oxford Entrance Examination on the basis of an interview and school reference. The scheme is described at https://www.hertford.ox.ac.uk/alumni/hertford-today/neil-tanner-and-the-tanner-scheme/the-tanner-scheme. According to this, Neil Tanner had been told by headmasters that their students could not afford extra tutoring for the Entrance Examination (which indicates that such tutoring was felt to be desirable.). Some comments at the webpage https://www.hertford.ox.ac.uk/alumni/hertford-today/tanner/research-project also indicate that teaching for the Examination was thought to be desirable. This is of course subjective opinion, but there is also objective data to support this position. Neil Tanner’s speech from the 1970s about the ‘Hertford Scheme’ is available in a link from that page, and states that around a third of applicants who entered through that scheme ended up with a First, while at that time the overall figure was around ten per cent.
This indicates that in terms of raw ability those who gained admission through the Hertford Scheme (who by definition were unwilling to take on the Entrance Examination) were, on average, superior to those who gained admission through the Examination. I think Tanner’s point here is that is that the Scheme was a better indication of raw ability than the Examination. This explanation for this may be partly because the Examination was contaminated as an indicator of ability by the fact that some people were better prepared for it than others, and partly because the Scheme might have been a better indicator of potential than even an Examination taken under the ideal (but unattainable in practice) conditions in which every candidate was equally well prepared. The reasons are immaterial for my argument : my point is that even this exceptional cohort of applicants (those who got in through the Scheme) were unwilling to take on the Entrance Examination, and thus however good (or bad) the Examination might have been as a test of potential and a predictor of future achievement, taking it was much harder for those without special preparation.
The Hertford website also has a historical section devoted to the stories of those who benefited from the Tanner Scheme at https://www.hertford.ox.ac.uk/alumni/meet-the-hertfordians/50-years-of-access, and, to indicate where we are today, a page describing a current mentoring scheme :https://www.hertford.ox.ac.uk/alumni/publications/hertford-blogs/mentoring-opens-doors-to-oxbridge
The opinion that fourth term applicants were disadvantaged compared to fourth term applicants is expressed at https://dynevorrevisited.org.uk/in-memoriam-robin-chapman-1976-1981/
Other references which may be of interest:
http://www.buckingham.ac.uk/wp-content/uploads/2012/07/Educating-the-Highly-Able-Report.pdf. See section 7.37 "Exceptionally Able"
http://demorgandotorg.files.wordpress.com/2012/02/gardiner_able_10aug12.pdf. Note the comment about the need for special provision for the the exceptionally able
https://royalsociety.org/topics-policy/publications/2012/raising-the-bar-developing-able-young-mathematicians/
https://www.undergraduate.study.cam.ac.uk/applying/contextual-data
https://www.maths.ox.ac.uk/study-here/undergraduate-study/maths-admissions-test/mat-livestream
https://warwick.ac.uk/fac/sci/maths/admissions/ug/june2018-version2.pdf
https://web.archive.org/web/20150909201429/http://occamstypewriter.org/athenedonald/2014/10/24/cambridge-admissions-dispelling-the-myths/
https://amsp.org.uk/students/university-admission-tests/step-mat-tmua
https://www.theguardian.com/education/2011/jun/20/oxbridge-access-poor-students
https://www.christs.cam.ac.uk/how-apply-1/low-participation
https://www2.physics.ox.ac.uk/events/2020/06/27/preparing-for-the-pat-2020
http://www.ctc.cam.ac.uk/news/190904_newsitem.php
https://www.murrayedwards.cam.ac.uk/step-summer-school