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Looking across many different dictionaries and websites, they have different pronunciations for the word arbitrary.
A colleague says it is pronounced like ahr-bitri, but this seems incorrect to me.
I am of the impression it should be pronounced as ar-bi-trery. Whereby, I am pronouncing all the R's and the er as a short u.
Thus, what is the correct way to say arbitrary in British English and why?
Whenever you're trying to find out how to pronounce a word, a good first step is to look at the pronunciations that are listed in dictionaries. For the word arbitrary, we see that Collins does indeed list a pronunciation with three syllables ("ɑːʳbɪtri"; there are some things about this transcription that I wouldn't choose to include, like the superscript "ʳ" that doesn't correspond to any sound at all in a typical southern British English accent). Oxford "Living" Dictionaries also lists this pronunciation: it transcribes the word as "/ˈɑːbɪt(rə)ri/", where the parentheses indicate an optional part of the pronunciation.
There is no technical definition of "correct" pronunciation, so you're free to disregard these dictionaries, but I think that would be a bit eccentric. Most English speakers treat dictionaries as reliable sources of information about "correct" English. The peevers or sticklers who feel that contemporary English dictionaries are too "permissive" don't necessarily all agree with each other, so if you're interested in that kind of person's definition of "correctness", you'll have to decide on which particular peevers to accept as authorities.
In general, the ending -ary in British English is often syncopated to /ri/. This is less common in American English, where -ary is often pronounced with an unreduced vowel. Since abitrary has another /r/ sound before the ending -ary, the two /r/s combine (or one is lost) in the syncopated pronunciation /ˈɑːbɪtri/.
Syncope of word-internal schwa is common in various contexts in English words. Some speakers might call it "incorrect" based on some idea that "good pronunciation" is as close to spelling as possible, but this is far from a unanimous opinion. A phonetician or a good EFL teacher would be unlikely to hold this opinion, as schwa syncope is an observable fact of English as it is spoken by educated native speakers of standard dialects (the type of English that is typically considered the most desirable for a non-native speaker, or a native speaker of a "non-standard" dialect, to acquire—I'm simplifying things here, since there is actually no single "standard" of English pronunciation, or even any single standard of British, American etc. pronunciation, but I don't think the simplification makes a difference to the point that I'm making).
