I'm not not sure if it's an American/British thing
It is an American/British thing, although there also could be variation between individuals of either accent.
In general, British English speakers are more likely than American English speakers to elide a syllable in words ending in -ory (like repository) or -ary (like dictionary). When British English speakers don't elide the second-to-last syllable of words ending in -ory/-ary, they generally pronounce it with a reduced vowel (also known as schwa, and transcribed as /ə/). The same reduced vowel sound is found in either accent in the second-to-last syllable of surgery or of battery, when that word is pronounced with three syllables.
American English speakers are more likely than British English speakers to use a pronunciation with an unreduced vowel (what you wrote as "reposi-to-ree"). The syllable containing the unreduced vowel can be analyzed as having some stress, although not as much as the syllable with the "primary stress" of the word (in repository, that is the second syllable). This "minor" kind of stress has been called "tertiary stress". Tertiary stress usually isn't present when the immediately preceding syllable is stressed (tchrist left a comment listing some words like this, e.g. history, which in both accents typically has no stress of any kind on any syllable other than the first).
There is a question on ELL asking about why American English and British English differ in this way, but I don't think the reason is entirely clear: '-…ory' : Pronunciation difference between American and British English?
or do people just use different pronunciations on special occasions.
I haven't heard of people varying the number of syllables in the word repository based on the occasion. Some people might have what is called "free variation", where they use both forms more or less interchangeably. As far as I know, neither pronunciation is considered markedly informal, so I wouldn't worry about one or the other pronunciation being inappropriate for any situation. At most, using the less common pronunciation for your environment might sound unusual, but I don't think it would be a big issue.