A number of the citations in the OED from the 18th and early-19th century have the word spelt rout, which leads me to suspect that the word was routinely pronounced to rhyme with out in Rightpondia during that period. Presumably in Britain the pronunciation was ‘corrected’ to conform with contemporary French during the Victorian period. This was a distinct trend; around that time venerable pronunciations of names and words such as ‘Eddard’, ‘Annis’ and ‘cowcumber’ were deemed sloppy and ignorant, and tidied up to reflect their spelling - Edward, Agnes and cucumber.
I don’t know how long the out pronunciation hung on in educated British English. In a set of lyrics for the regimental march of the Connaught Rangers written in 1890 by an officer of the regiment, route is made to rhyme with doubt; but that may just be for Irishism - it also has tea rhyming with way, leap with step, easy with crazy, etc. Also, it’s quite possible that the military could have retained the old pronunciation as a technical term long after it had been abandoned in general British use.
Meanwhile, presumably the Americans went on saying it as they always had done. The OED suggests that the use of ’Route [numeral]’ to designate a major highway was influenced by the numbered French Routes nationales, which would account for the adoption of the French pronunciation.