The OED pronunciation of primer, n.1 is
(in sense 2d) Brit.
That shows that apart from sense 2d, the ‘long i’ version is preferred in Britain and the ‘short i’ version is preferred in the U.S., and that apparently everyone says the ‘short i’ version for sense 2d (see below).
It also notes that:
Pronunciation with ‘short’ i (
/ɪ/) is original (and is still usual in senses relating to type); pronunciation (in the other senses) with ‘long’ i (now
/ʌɪ/) seems to be first recorded in British dictionaries of the late 19th cent. and is the primary one given in all editions of D. Jones Eng. Pronouncing Dict.
Which says that the ‘long i’ version is new, and that the original and historic pronunciation is the ‘short i’ version, which America has retained more than Britain has.
The referenced sense 2d is
Chiefly N.Z. A class covering one of the first years of instruction in a primary school; a child in a primary school class.
Third edition, June 2007; online version March 2012. http://www.oed.com/view/Entry/151307; accessed 08 May 2012. An entry for this word was first included in New English Dictionary, 1908.