Thanks, I think it's great to make this data available and to discuss it.

FWIW, while I haven't looked at any updates the UN may have made for this iteration, when briefly comparing the previous UN projections with those by Vollset et al. (2020), available online here, I came away being more convinced by the latter. (I think I first heard about them from Leopold Aschenbrenner.) They tend to predict more rapidly falling fertility rates, with world population peaking well before the end of the century and then declining.

The key difference in methods is that Vollset et al. model fertility as partly depending on educational attainment and access to contraceptives. By contrast, I believe the UN (at least in the previous iteration) primarily did a brute extrapolation of fertility trends. 

Assume we believe the causal claim that increased educational attainment and access to contraceptives causes lower fertility. Then modeling these two causes explicitly will beat a brute extrapolation of fertility if there are countries which:

• Haven't seen much of an increase in educational attainment or access to contraceptives, but will do so in the future; or
• Have seen an increase in these two indicators which, however, (because of time lags or threshold effects) is not yet visible in falling fertility rates.

And I think this happens to apply to quite a few countries.

Some caveats:

• I only had a pretty quick look at both the UN and the Vollset et al. projections. I wouldn't be that surprised if my above summary of their respective methods is based on a misreading on my part or otherwise inaccurate in important ways.
• There are obviously causal drivers of fertility that aren't modeled by Vollset et al. either, such as policy effects – i.e., we might reasonably expect that countries will try to incentivize higher fertility (e.g. tax benefits for households with more children) once they start bearing significant costs from shrinking populations that are increasingly dominated by the elderly. (Similarly their model assumes that, e.g., there won't be technological developments such as transformative AI or perhaps more mundanely things like artificial wombs that change which sorts of things fertility causally depends on.) So arguably both the UN and Vollset et al. should best be viewed as projections holding non-modeled variables constant as opposed to all-things-considered predictions.
• An indirect and in my view fairly weak reason for skepticism about the Vollset et al. projections is that they are by a team of researchers at the Institute for Health Metrics of Evaluation (IHME), and that other work out of IHME was controversial in several instances, including their early COVID models. (I haven't looked into how valid the criticisms of their other work are, nor whether there is any overlap in researchers between previous criticized work and their population projects – at least the first authors/lead researchers seem to be different.)

I suspect there's a good chance that populations in Western nations could be significantly higher than predicted according to your link. The reason for this is that we should expect natural selection to select for whatever traits maximize fertility in the modern environment, such as higher religiosity. This will likely lead to fertility rates rebounding in the next several generations. The sorts of people who aren't reproducing in the modern environment are being weeded out of the gene pool, and we are likely undergoing selection pressure for "breeders" with a strong instinctive desire to have as many biological children as possible. Certain religious groups, like the Old Order Amish, Hutterites, and Haredim are also growing exponentially, and will likely be demographically dominant in the future.