One thing about fossil fuels is that they have been isolated from the atmosphere for a very long time. That means that the carbon-14 produced by cosmic ray impacts with nitrogen nuclei in the atmosphere that they originally contained has pretty much decayed away. So, when fossil fuels are burned, carbon-14 in the atmosphere is diluted with respect to carbon-12 in the atmosphere. But that diluted mix is just where our food comes from, and since we are what we eat, we end up with less carbon-14 in our bodies. If we were to instantaneously double the concentration of carbon dioxide in the atmosphere by burning fossil fuels, the amount of carbon-14 in our bodies would go down by half. This would cut the beta decay rate from carbon-14 in our bodies by half and the combined beta decay rate from potasium-40 and carbon-14 by about 12%. If we subsequently removed carbon dioxide from the atmosphere to control the climate, carbon-14 would be removed together with carbon-12 and the internal radiation burden in our bodies would remain low for thousands of years as carbon-14 production would catch up on a timescale similar to the decay timescale for carbon-14 (5700 years).
Now, potasium-40 decays are more energetic than carbon-14 decays but carbon-14 in incorporated in DNA and thus decays may have a greater chance of inducing mutations, the apparent origin of cancers.
Nuclear power use, on the other hand, only increases radiation exposure, it does not decrease it. And it does so for long long after there has been any benefit from the power generation. While there are immediate cancer risks in the chemical components emitted from coal, gasoline or diesel burning, those risks are incurred by those benefiting from the fuel use and are not transmitted down the ages. The long term cancer legacy, in fact, is reduced cancer risk owing to reduced internal radiation burden.
A review of carbon-14's especially damaging roll in DNA may be found here .