English is not a language known for its consistency in pronunciation. To a speaker of a language such as German it must seem odd that rough, bough, and cough have a completely different sound. Given this variety it would be churlish to worry about American pronunciation of words like tomato and basil. I will even allow for Aluminium being pronounced as if it ended in –num and not nium; as this is how it is spelt here in the US. There is one word that I draw the line at; route is a homonym for root and not for rout.
The classic American route is Route 66 that ran from the Chicago to Los Angeles. The road was immortalised in the eponymous song by Bobby Troup. If you play the Nat King Cole performance below you can clearly hear the correct pronunciation.
A Brit in California 
You know, I am honestly quite relieved that I am not the only one pedantic about pronunciation. That said my spelling is till shocking and find myself pretending to be American when I get it wrong… works well 😀
As a Californian, I am exhausted with attempting to correct Brit’s spelling and pronounciation. We had the ‘root’ – ‘rowt’ discussion last week concerning the word ‘router’, and I graciously allowed for the fact that he is a foreigner just learning the ropes here in the States, and that there is a certain quaint charm to the determined mispronounciation of certain words. However, perhaps I was too accomodating …
You would have to do more than correct the English, the French and the Romans too. “Route”, circa 1225, from the Old French “rute”, from the Latin “rupta”.