Dueling Movies: With An American Tail, another animated movie about mice coming out the same year (though it didn't come out until a few months afterward). The Great Mouse Detective lost in the end, but that didn't mean it wasn't successful; in fact, it convinced Disney that its animation unit still had a future.
The success of An American Tail probably helped convince Disney as well.
Executive Meddling: Largely averted, as Jeffery Katzenberg had little to no faith the movie would be any good. Most of his attention was spent on meddling with The Black Cauldron; we all know how that film turned out. Katzenberg did, however, demand the title of this film be changed from "Basil of Baker Street" to simply "The Great Mouse Detective". Two of the four directors have since said they've always hated this new title.
However those directors may have had some solace in the title being changed to "Basil the Great Mouse Detective" is some countries, including Australia.