Usually rowt, but I believe I say root when I mention Route 66. When I plan a trip I am choosing the rowt though. Also, addresses such at Rt. 55 Box 19 is rowt.
One reference listed 3 pronunciations---American, British, and Canadian, but I didn't pursue it. To some degree it's nomenclature in context rather than a defined pronunciation.
In standard British English, it's "root". But as for the magic box that allows me to communicate with the rest of the AM community . . . that will always be a "rowter". Inconsistent, to be sure. But inconsistency is my middle name.