This is simply unrelated. Fossil fuels are worth a lot of money and are frequently located in areas controlled by unstable regimes, hence wars. There are also wars over water, diamonds, gold, arable land, and any other high-value commodity located where the local political environment is unstable.

Using a lot of nuclear power would not alter this picture in the slightest. It might alter the landscape in terms of supply and demand, perhaps shifting fossil fuel demand from heavy nuclear users toward developing or lower-tech countries, but the basic equation of high value resources plus political instability remains intact.

California is highly oil-dependent and Texas has a lot of oil. So far California has not invaded Texas. That's because both are politically stable.