'Fox Family' had first shot at this -- in 2000!
Believe it or not, Fox Family Channel (the cable network now known as ABC Family) had a chance to broadcast this show under its literal English title, "Detective Conan", back in 2000. It looked as if everything was set, and Fox Family even advertised that it would be part of their "Fox Kids" lineup.
Well, Fall 2000 arrived, and Detective Conan was nowhere to be seen. From what I've been able to dig up on the 'net, in February of 2001 it was announced that FF had never received the rights to air the show and the plans to air it had been scrapped. The official reason given was that there was an issue with the violence on the show.
One source - not an official or authoritative source, mind you - claims that Fox still owns the North American copyrights to the show title "Detective Conan" and the character names. That is supposedly why the show had to be renamed "Case Closed" and Shinichi Kudo is now Jimmy Kudo, Ran Mouri is now Rachel Moore, etc. The same source claims that Conan Edogawa is still Conan Edogawa because the "Conan the Barbarian" people own the rights to the name Conan and Fox couldn't lay claim to it. (Sounds kind of flaky to me...)