The word route has two common pronunciations, namely [ruːt] and [raʊt]; you can listen to both forms here. However, it is not so clear which version is used by whom and in which context. In particular, it is not a simple matter of British vs. American English.
According to Longman’s Pronunciation Dictionary, [ruːt] is used for almost all purposes in British English, except in the military context where [raʊt] is used. In American English, [raʊt] is more common. A good example is the famous Route 66, which is usually pronounced as [ru:t] in songs and movies. A survey conducted for Longman’s Pronunciation Dictionary reveals that roughly two thirds of all Americans favour [ru:t] whereas one third prefers [raʊt].