Word histories are often characterized by twists and turns. A good example is router, which is derived from the word route (of Anglo-Norman provenience, i.e., Middle English < Old French < Latin). In contemporary American English the alternate form of the deriving base [raʊd], rhyming with rout instead of root, clearly stems from a reading (= spelling) pronunciation and is still typically listed second in the dictionaries.

Of course, anyone who knows the song “Route 66” (lyrics by Bobby Troup) will not fail to give the word route in the refrain its proper “British” pronunciation, as in:

If you ever plan to motor west,
Travel my way, take the highway that is best.
Get your kicks on route sixty-six.

The persistence of [ruːt] (and the total inappropriateness of [raʊd]) here is to be explained by the poetic design in its phonic aspect: the internal rhyme kicks/sixty-six utilizes the high unrounded vowel /i/, which dictates the presence of the corresponding high rounded vowel /u/ in route.

By contrast, the new meaning of router (it has several older ones) connected with internet technology is unexceptionally pronounced [ˈraʊdər] on this side of the Atlantic. Here is its complete entry from the Oxford English Dictionary Online:

router, n.
Brit. /ˈruːtə/ , U.S. /ˈraʊdər/
< route v. + -er suffix1.
and Computing.

A device, circuit, algorithm, etc., which serves to determine the destinations of individual incoming signals; esp. a device which receives data packets and forwards them to the appropriate computer network or part of a network.

1968    Nucl. Physics A. 116 549   A router circuit sent the coincidences from the first unit to be stored in the first 200 channels of the pulse-height analyser and those from the second to the last 200 channels.
1970    Nucl. Instruments & Methods 85 64/2   A ‘router’ switched the output of the detector to each of the subgroup in succession.
1986    Science 28 Feb. 976/2   The router can pick a component of the node address that is not zero and send the message in a direction in which that component of the node address is one.
1990    Pract. Computing Sept. 85/3   This enables printers with Apple’s built-in network, Localtalk, to be connected to Ethernet‥without the need for an expensive gateway or router.
2006    Hi Life No. 5. 34/1   If you add a Wi-Fi router to your broadband link you’ll be able to access the internet via Wi-Fi-equipped laptop from any room in your home.

The explanation in the case of the derived word is its MARKED STATUS, i. e., an agentive in –er that is an object, not a person, hence conducing to the iconic pronunciation (= word and meaning forming a diagram in the semiotic sense) with the marked vowel /aw/. This is a good illustration of markedness agreement (between sound and sense) being a definitive––if only potential–– telos of language change, not a necessary one. British English, by contrast with American, has not yet exploited the semiotic potential inherent in this particular case of lexical development.