Learn English – Meaning of “take over”

Ama and her daemon climbed up over the rock shelves and around the little cataracts, past the whirlpools and through the spectrum-tinted spray...

says that Ama and her daemon climbed up, and then in nice parallelism tells where they went or what they encountered on the way:

• over the rock shelves
• around the little cataracts
• past the whirlpools
• through the spectrum-tinted spray

Thus, while up over may be slightly pleonastic, it is useful as part of a rhetorical device to better evoke an emotional response in readers.

There is no significant definitional difference between the three expressions, but they differ in connotation, so they cannot exactly be used interchangeably either.

"To take a role" is a neutral expression, which can be used in a variety of ways but usually serves as a denotation that implies nothing else. It doesn't seem like what you're looking for.

"To take over a role" is, as you said, to take a role that was previously occupied by somebody else, which seems to be what you're looking for, as your colleague "replaced" another worker, so he "took over" that worker's role.

"To take on a role" is, again as you said, to accept the role as a challenge. However, you seem to be emphasizing the replacement aspect more than the challenge aspect, at least from what I can tell by your description.

I'd personally recommend "take over the role".