Americans traditionally “took” tests or exams. Now, increasingly, they are sitting. What’s next: standing for a by-election?
Certain readers of ”Herzog” complained the book was difficult. Much as they might have sympathized with the unhappy and comical history professor, they were occasionally put off by his long and erudite letters. Some felt that they were being asked to sit for a difficult exam in a survey course in intellectual history. (Saul Bellow, New York Times, March 8, 1987)/City School Superintendent Neel Durbin opened the meeting by celebrating the number of Dyersburg High School students enrolled in AP courses as well as the number that are passing the AP exams at the end of the course. …”Fifty-five percent of our students that sit for the exam pass it,” said Durbin… (Dyersburg [PA] State Gazette, October 18, 2011)
notoneoffbritishisms.com 
Britains sit an exam or sit the exam, they don’t sit for exams – so you can’t blame us for this one!!!
We sit for our portrait but sit an exam.
The OED lists all these (British) uses of “sit for,” 1929-1980:
1929 R. Graves Good-bye to all That xxvii. 362 My tutor‥warned me that I must on no account disparage the eighteenth century when I sat for my final examination.
1955 Times 30 June 6/5 Pupils sitting for the examination for entry to secondary schools.
1963 R. Pedley Comprehensive School i. 14 In some of the 3900 ‘modern’ schools in England and Wales it is possible for the cleverer pupils to sit for GCE at ordinary level.
1968 G. Maxwell Raven seek thy Brother viii. 102, I appealed to my guardian to be allowed to retire‥from the scene on the grounds of ill-health.‥ The refusal was absolute‥; I was to sit for my degree, no matter what the outcome.
1980 Radio Times 1 Mar. 16/4 It is possible to take an A-level without having sat for the O-level.
Voila the explanation: with the exception of Gavin Maxwell, the citations are all London/southern England based and, in the later cases, the references are to sitting for the qualification not the process, i.e. the exam / test. To balance the argument, how many citations are there in the OED for ‘sit an’ or ‘sit the’ for the same period?
Scrolling down the OED, I find these (the first time through, I stopped at “sit for”):
1957 A. Wilson Bit off Map 40 With the degree behind me, I shall sit the Administrative in June.
1966 Rep. Comm. Inq. Univ. Oxf. II. 152 Collections are college examinations, usually sat at the beginning of a term.
1980 Radio Times 1 Mar. 16/4 A child can‥enter for and sit an examination without being put forward by the school.
No U.S. presence, and I don’t expect to see one!
When I was living in London (1960s-80s), it was both ‘sit an exam’ and ‘sit for an exam’ all the time. We hardly ever heard ‘take an exam’ until the mid-1980s – and then usually from American or Canadian exchange students.
The answer to my earlier coment is possibly provided by Edward T Hall who tells us that language reflects culture. The UK is, by his definition, a high context culture so not everything has to be said and ‘sit the exam for my degree’ becomes ‘sit for my degree’ as everyone knows that it involves an exam. The USA is a low context culture where such economy of words is inappropriate and communication has to be more explicit. This appears to be confirmed by the US (mis)appropriation of sit FOR an exam but that does not answer the initial question of sit or take…
I most certainly can attest to the UK being a high context culture, having lived in both the UK and the USA during my formative years, plus a few other countries. Indeed, in my own experience, the USA has been the only place where everything has to be spelled out to thy kingdom come – and that’s seriously tiring and frustrating on a daily basis. Other people who’ve lived in the USA and other places also concur with me, broadly speaking. The only other ‘group’ with the same spell-everything-out characteristic is the Chinese – and I go through spelling out the bejesus of everything every day now that I’m living in Hong Kong.
I noticed there is no entry for a very distinctive Britishism that I have never heard in the U.S. – “set an exam” meaning “give/administer an exam.” In one of David Lodge’s novels, the phrase is used by an American character – even skilled writers can sometimes not realize what phrases only exist on their side of the Atlantic.
In some parts of Quebec, Canada “sit” has always been the normal verb for taking a test or exam. In Montreal an exam is a normal “test”, whereas elsewhere in Canada an exam is only a term- or year -“final” test, quiz being the shortest and least weighted of the three. I call it “borderline” because of Quebec, but everyone in Eastern Canada says “take/write an exam/test”.