Those numbers simply don't make sense for several reasons. They are with subsidies (not without) as the web page itself states. The numbers for nuclear are likely based on the failed European projects as China and Russia are building a lot of new nuclear power plants cheaper than that.
The biggest issue, though, is LCOE completely ignores the costs associated with the intermittency of wind and solar power. The LCOE is artifically low for intermittent sources and vice-versa for baseload power sources (coal, nuclear, hydro, etc.). When the conditions are right, wind and solar produce a lot of power, but this is not correlated with power demand, and the market price per kWh goes down so much during that time that baseload power is not profitable. Baseload power production is reduced, but the total installed base must remain the same (because you need power when the sun doesn't shine), so LCOE goes up.
LCOE is, conversely, too low for intermittent sources because it only cares about total accumulated production, not if the power was produced when it was actually needed.
I don't know what you're reading but it clearly states unsubsidized wind in the second graph (which has the same value as the first graph).
Regarding subsididies while it does not receive the most subsidies now nuclear has received 66% of the R&D and demonstration subsidies from 1974-2007 in the EU. It still receives about half of what solar receives [1]. So despite it having received much more subsidies previously its still not viable.
I'm sorry, I have misread your source. It clearly states "unsubsidized". My bad.
I think my points about the LCOE still stand, though. The LCOE is not a good measurement for comparing intermittent and baseload power sources. Nuclear in particular suffers here since the capital costs are very high compared to operating costs. A nuclear power plant needs to run close to maximum power all the time for maximum profitability. This artifically inflates the numbers for nuclear while lowering the apparent cost for intermittent sources.
Regarding your second paragraph, I'm not sure if you're arguing that the EU spends half the amount on nuclear power R&D as solar power R&D, or if you're talking about subsidies for the plants themselves. I'd argue grants for R&D projects would be very hard to count in a meaningful way, and I don't think e.g. a research grant that results in more efficient solar voltaic cells should count as a subsidy towards solar power, nor nuclear fission research as a subsidy towards nuclear power.
The EU provides grants for constructing e.g. wind power farms. Unfortunately, I'm not familiar with every country, but in Sweden, renewables are given a large cashback grant per kWh generated ("elcertifikat"). Nuclear power here has been profitable since the 70s, and didn't turn unprofitable until taxes on nuclear power were increased substantially a few years ago.
The biggest issue, though, is LCOE completely ignores the costs associated with the intermittency of wind and solar power. The LCOE is artifically low for intermittent sources and vice-versa for baseload power sources (coal, nuclear, hydro, etc.). When the conditions are right, wind and solar produce a lot of power, but this is not correlated with power demand, and the market price per kWh goes down so much during that time that baseload power is not profitable. Baseload power production is reduced, but the total installed base must remain the same (because you need power when the sun doesn't shine), so LCOE goes up.
LCOE is, conversely, too low for intermittent sources because it only cares about total accumulated production, not if the power was produced when it was actually needed.