Honestly quite a bit, but, unfortunately the response time of Nuclear power is not the fastest available. It takes hours to get the process going, or to shut it down, and unfortunately an energy grid needs to have a demand-supply net of 0 (usually plus/minus 1.5%) or the appliances connected might face some irreversible damage. This means that, while Nuclear could have been a great answer to a good chunk of the energy demand (sorry cannot estimate how much), we would have probably still needed a lot of the traditional fossil fuel source fo generation.

The other aspect to always keep in mind, when looking at a system as mature as the energy grid is COSTS. Most likely the economy of nuclear doesn't work out compared to the alternatives available. (that's why policies and incentives are needed to move away from fossile fuel)