Is the English language used by the European institutions the British one?

I am working for an international organisation which includes most of Europe, but also some other major Anglophone countries (most notably the US). Although effectively all communication happens in English (and de facto in AmE), if one were to propose the actual adoption of English as a lingua franca within the organisation, countries like France and Germany might (threaten to) quit the organisation, and the mere mention of "one lingua franca" might spell the end of existence of the host country (Belgium).

Now, within the EU, language is a very fickle point, and it should always be approached with extreme caution. The list of languages into which every official EU document has to be translated is longer than one would expect, and I am quite certain that the majority of EU citizens outside the UK and Ireland would not expect to (have to) read an EU document in English.

If you are really adventurous, travel to France and propose the idea that the French representatives to the EU are a target audience for the English publications.

Warning that adventurous proposal is not meant to be actually executed!

Edit: sneaking a peek at your profile I notice you are in France already. You might want to give the idea a try - just be very careful. Interestingly enough, even if you would convince France (and I doubt it!), there are plenty of other countries that will refuse to give up their right to be addressed in their own language. The policy of the EU is to accommodate for those wishes, and thus, the English versions of documents will mostly be read by the people in the UK and Ireland.

Hence, it is likely that the form of English used in those publications will indeed (largely) follow the rules of British English.

I work for various educational institutions in Germany and, therefore, within the EU. I can confirm that British English is the preferred variety of English used within the 28 EU member states.

When using the European Commission's website, http://ec.europa.eu/translation/english/english_en.htm, the translation information into English corresponds exclusively to the British culture, rather than to the American culture.

Using the IATE, the terminology database of the European Union,

http://iate.europa.eu/SearchByQueryLoad.do?method=load

you will also receive answers corresponding to the British culture. For example, if you want to translate the German word "Gymnasium" into English, you get "grammar school". In the UK, "grammar school" is the equivalent of "Gymnasium" in Germany, which is a college-preparatory high school. But in the US "grammar school" is another name for "elementary school" or "primary school". This can be very confusing.

The list goes on and on.

It is understandable that the European Union uses the variety of English that its members use, which is the English of the UK and Ireland. When it comes to international business, though, North American English is much more common.