Given sufficient time small animals would develop gigantism, but not on the timescale that you are asking.
Gigantism is a frequent occurence in evolutionary history, not just on islands. Usually what occurs is the dominant large species are wiped out and lineages of smaller, more adaptable species become gigantic and take their place. Generally this happens because bigger species are more specialized and sensitive to environmental changes and generally exist at lower population sizes. This is one form of what is known as a macroevolutionary ratchet.
The most famous example of this kind of thing is the K-T extinction. Big dinosaurs die, and within a few hundred thousand years you have mammals tripling in size and eventually taking their place. This also happens a lot within groups as well. Tyrannosaurs were dog-sized carnivores under foot of allosaurs and megalosaurs in the Jurassic, and modern dogs, cats, and ruminating ungulates were small, inconspicuous creatures until the Miocene, whereas before then those niches were occupied by odd-toed ungulates, bone-crusher dogs, and nimravids, among others.
If you have a mass cataclysm enough to wipe out most of humanity, it's going to take most of the dominant megafauna with them. Depending on how severe it is you may have something on par with the K-T, where only housecat-sized or smaller animals survive, which would mean complete extinction of things like odd-toed ungulates (i.e., horses), most even-toed ungulates (at best maybe some small duikers survive), and most carnivores. At that point you would get the surviving species adapting to fill the empty niches. Maybe shrews, rats, and weasels become large carnivores, and rodents and rabbits become the new herbivorous megafauna.
However, three important things to note...
- This will not happen within a century. One century from doomsday you are going to have a bunch of small mammals scurrying over the ruins of the modern world. It takes thousands of years for nature to recover from a mass extinction. It took 700,000 years for mammals to become pig or wolf-sized after the K-T. If one century were all it took for gigantism to develop you would see species becoming gigantic every time there was a local collapse of civilization throughout human history.
- You are not going to have everything become gigantic. There are still going to be small rats and shrews under foot of this new megafauna, waiting for this batch to go extinct so it is their turn. Small members of a clade can die out, but even in that case something is going to fill the niche.
- What animals do evolve will likely not be identical to scaled up shrews, rats, etc. Big animals have different structural and ecological problems than small ones, and so their anatomy is going to reflect that. They will look shrew or rat-like, but they will have their own specializations and adaptations for large size (or feeding on large prey, etc.) and you cannot simply say "this is a Rattus norvegicus the size of a grizzly". For example shrews lack a zygomatic arch because at their small size they don't need it, but at larger sizes it becomes useful for buttressing the skull when biting. They might end up re-evolving a zygomatic arch similar to golden moles, or develop a complex jugal similar to ground sloths. They won't look like modern shrews scaled up.