The rule is not hard and fast, however. Some BrE speakers will use "root" for "path travelled", and I've heard both BrE and AmE speakers describe their spinning bladed power tool as a "rooter".
Maybe writing "router and router" wasn't very clear. Those are "rout-er" and "route-r".
As far as I know (and I'm British), for path travelled or a device directing those paths, all BrE speakers rhyme route and router with hoot and hooter.
The power tool, the older manual tool, and the slots they cut, are more obscure. Rout and router should rhyme with shout and shouter in BrE, but I would expect mistakes or guesses for this router.
With shoot/shout, boot/bout, loot/lout, scoot/scout, snoot/snout, toot/tout, in BrE a path travelled should be a root. But there's also brute/flute/jute/lute etc, which could be better written broot/floot/joot/loot, leaving the -ute for words like mute, tribute, minute (small), repute.
The same difference holds for router and router.