Your argument assumes we already have nuclear energy, which, by and large, we do not.
1. This means we would have to build new power stations, which assumes centralised distribution, which means building those power stations somewhere near the existing coal fired stations - because that is the way the distribution network is designed. Coal fired stations are located near sources of coal - not on sites which might be good for nuclear reactors. E.g. near rivers.
2. This means continuing to maintain and upgrade the existing distribution network, which the public are unwilling to do, because they can generate their own power on their own roof, and don't see why they should subsidize industry by paying the bill for the centralised network.
3. Nuclear power is heavily IP bound, we would have to buy technology from, for example the US or other pre-existing user of nuclear power. Again, this looks to the public like money down the drain.
4. Depending on who you are, acquiring nuclear technology can make the legacy powers (e.g. the US, UK) nervous. Who needs those guys on your back?
5. Even if you have uranium (we have in my country) you still can't just feed it into your reactor. Many reactor designs require you to have the fuel rods made overseas. Again, this looks bad. Why are we adopting a source of power that makes us reliant on overseas companies?
If one of the newer designs was (a) ready to go and (b) commercially viable and (c) open source, so you could build it yourself (d) able to take locally sourced fuel so that you don't have to ship things high and low and (e) relatively foolproof, then in manys case nuclear might be viable.
But not everywhere. There is no way nuclear is suitable for Nigeria, for example. What happens when Boko Haram gets their hands on a nuclear power site? In many places, the key to energy generation is finding ways to distribute the generation so that it generated near the site of consumption. This way, you can have, for example, community owned generating facilities.