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I presume that toroidial body is magnatized and rotating I would expect that the magnetic field would remain confined to this core and very little would leak out to generate current in the coil.This appears to be half of a torordial transformer one of the virtues of which if little stray field.
If a spherical magnet was rolling around inside the hollow toroid I believe a current would be generated, (need more coils though) ... ...sorry, you cannot view external links. To see them, please
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Does the magnetized object have to be in the form of a toroid ? for your thought experiment would not a rod of quasi infinite lenght be better ?
In what manner would it be magnetized would it have poles at its ends or are the poles at either side of the slot ?.