Nuclear has a horrible failure rate. Of around 435 civilian reactors in the world, 6 have failed catastrophically which is about 1.3%. Many more have had serious but less costly failures and accidents. Imagine if 1.3% of passenger aircraft crashed, it would never be tolerated. Energy seems to be an exception though, e.g. we tolerate massive oil spills and wars over it.
It's a case of there being no better alternative.
I am pro wind, and to a limited extent pro solar. However, they cannot provide for 100% of our demands without ridiculous amounts of grid storage. Simple fact - during the June/July months wind power was only providing about 10% of its nominal output. How can that work? We'd need 10 x as many turbines to just get a base load, and most of the time, they'd just be off or idling, making the cost of wind power far too high.
Nuclear is extremely safe compared to other options. The worst nuclear disaster, Chernobyl, has killed about 6,000 people. A series of design errors, poor staff training and poor maintenance lead to the disaster. But alone in the USA, coal power kills ~7,500 people per year due to pollution.
The problem with nuclear is when it does go wrong, it often goes really wrong. This makes for bad PR, nuclear has serious problems with PR. But it's comparatively very safe compared to the alternatives and it's an acceptable risk in my opinion. Flying in an aircraft is something like 100x safer than driving a car, yet you don't see 24/7 coverage of car accidents. We take flying and driving as acceptable risks, we must take nuclear as the same. (Coal could be considered an acceptable risk, but personally I think it's too much of a risk compared to the available alternatives like nuclear.)
Later reactor designs are extremely safe. Some fast reactor Gen 4 designs (prototype/research stage currently) can even work on the waste of Gen 3 reactors, which would close the fuel cycle. Of course the main reason nuclear plants aren't built as often as they were is NIMBYism. I'd honestly be 100x happier to have a nuclear plant near me than a coal plant.
Wind is cheaper than most nuclear power plants to operate, so should sell into the grid in periods of high wind, reducing overall nuclear demand and electricity prices. Since reactors take time to spin down, this would have to be planned in advance, or grid storage for the wind/nuclear would have to be used, but on a much smaller scale compared to pure solar/wind solutions. Solar can also sell into the grid but in the UK I don't really see much benefit, except maybe on residential houses as an investment.