The Great Mouse Detective is just a slice above the standard fare Disney was producing in the 70s and early 80s. Basil (the mouse version of Sherlock Holmes) is an amusing interpretation with fun errors and the occasional deep depression. Vincent Price plays the joyful and occasionally scary Rattigan, who wants to rule the mouse world.
Where does this world come from? It seems that The Rescuers and this film are all located within this strange world in which some animals (but not all) have created a parallel existence alongside humans. What kind of circumstances created this?
I think it all begins with Mrs. Frisby and the Rats of NIMH (or, the movie, The Secret of NIMH). Humans experimented on rats and gave them great intelligence, some of which were greater than humans. We see in that movie that rats passed this along to some mice, and even started interbreeding with them, creating intelligent children. Suppose they brought some other species into this, using the human experiments? The occasional bat (which they would think were normal because they seem so rat-like, but it didn't work out), some birds, and other animals. But some animals would have the intelligence-gene and some would not.
As for rats, because they were the first to gain intelligence, many of them would think that they should rule this animal world. However, the mice, because of their quick breeding, quickly took over and created underground societies. This led to many minor skirmishes between mice and rats, such as the one in this film.
BTW, I really prefer the title, "Basil of Baker Street". It flows so much easier.