Private funding implies loan rates which must ensure investor’s profit and commercial-driven discount rates. External costs (externalities) are typically ignored by commercial funding, unless enforced by regulation.
But the discount rate has a dramatic impact on the LCOE especially as these projects usually span decades. This is discussed in great detail in the latest NEA report (2020), Chapter 5, page 84 (chart taken from there).
The result of using private funding of long-term infrastructure projects - and it’s the same case for nuclear as for hydro or wind or PV farms - is that for most of their life they’re just paying out the interest, which can take up to 60% of the total cost project.