We in the US actually use both of these different pronunciations, one of which tends to be based upon the part of speech.
1) As a verb used for war (to route an enemy), we usually say "rowt" (rhymes with "out").
2) As a noun meaning "path", either "rowt" or "root" (rhymes with "moot" or "boot", etc.) is used.
• Some people might use one pronunciation for a path taken by a physical means of transportation (the route followed by a vehicle) and another for the path used by internet traffic (or a theoretical path).
=> The pronunciation choice for "path of internet traffic" usually dictates the pronunciation used for the related terms
- to route, verb referring to choosing the path, including exchanging routing information between...
- routers, noun referring to the specialized computers which PERFORM the routing (verb, last example).