"Rewiring" the brain isn't quite that simple: simply splicing a nerve to another doesn't necessarily mean the axons of that nerve will then grow into area. I am unaware of any studies that cross these specific pathways but as I am writing this answer I see you have already updated your question to ask about other modalities! So, with that in mind...
Crossing Auditory and Visual Pathways
Some related experiments have been done with other modalities: the ones I am aware of involve the auditory and visual systems. Both auditory and visual information traveling to cortex mostly comes through two nuclei in the thalamus: the medial geniculate nucleus, MGN, for auditory information, and the lateral geniculate nucleus, LGN, for visual information.
These thalamic nuclei are adjacent to each other. In Angelucci et al 1998 and Sharma et al 2000, lesions were made to cause retinal projections to innervate the part of thalamus that usually processes auditory information (medial geniculate nucleus, MGN). (note: there are also a bunch of older experiments in rodents and ferrets besides these, I'm not going to do a comprehensive review of the literature here but you can find many relevant citations within those other papers).
Briefly, the technique is to damage both the LGN and the pathways that normally project to MGN. The result is that the retinal axons, which normally synapse in the LGN, will instead path to the MGN. Note that it is important to do this very early in development, such experiments would not work in an adult brain.
The result, in the papers I cited earlier, is that the part of cortex that normally responds to auditory information begins to respond to visual information, and develops a hallmark of visual cortical response properties: selectivity to the orientation of presented stimuli. Therefore, it seems like if you give an auditory cortical area visual information, it doesn't "try to process it as sound" but rather processes it just like the visual cortex would.
Effects on perception
It is hard to ask the ferrets what their actual perceptions are, but the best guess is that perceptions are built by what we put into the system: there is no way for an area that receives anything but visual input to interpret that information as anything other than visual input. There is no context to interpret it any other way. Cortex develops and responds based on the information it is given. In the animals in these experiments, their auditory systems have been lesioned so there is really no way they could interpret the stimuli they "see" as "sounding like" something.
Similarly, if you send olfactory information to visual cortex, you will process smells in visual cortex. Olfactory processing is a bit different than the other modalities, so depending on where you took this information from things may not develop properly, but that gets into a lot of speculation that is beyond this .SE site.
Crossmodal plasticity in blind individuals
There are some experiments in humans that also look at this type of cortical plasticity. Probably the best ones are those in blind individuals. Again, I won't review the entire literature, but can give some examples: Cohen et al showed tactile representations in the visual cortex of blind individuals, such as when they are reading Braille. Bedny et al, 2015 show that, especially in children who are blind, the visual areas of the brain respond strongly to information from other modalities, especially spoken language. Again, these individuals are not "seeing" these stimuli, beyond the type of "seeing" that you can do when you reconstruct a scene with your eyes closed (if they were at one point sighted). Rather, part of their brain that would normally be used for vision is now being used for other tasks.
Effects on olfaction haven't been studied as often, but in Kupers et al, blind individuals were shown to have olfactory responses in brain areas that don't show olfactory responses in sighted individuals, including areas that usually process vision.
References:
Angelucci, A., Clascá, F., & Sur, M. (1998). Brainstem inputs to the ferret medial geniculate nucleus and the effect of early deafferentation on novel retinal projections to the auditory thalamus. The Journal of comparative neurology, 400(3), 417-439.
Bedny, M., Richardson, H., & Saxe, R. (2015). “Visual” cortex responds to spoken language in blind children. Journal of Neuroscience, 35(33), 11674-11681.
Cohen, L. G., Celnik, P., Pascual-Leone, A., Corwell, B., Faiz, L., Dambrosia, J., ... & Hallett, M. (1997). Functional relevance of cross-modal plasticity in blind humans. Nature, 389(6647), 180-183.
Kupers, R., Beaulieu-Lefebvre, M., Schneider, F. C., Kassuba, T., Paulson, O. B., Siebner, H. R., & Ptito, M. (2011). Neural correlates of olfactory processing in congenital blindness. Neuropsychologia, 49(7), 2037-2044.
Sharma, J., Angelucci, A., & Sur, M. (2000). Induction of visual orientation modules in auditory cortex. Nature, 404(6780), 841-847.
