The OP says:
" ...Recently, Palais wrote about it in the Notices but he only treats the old story from the 1950s and seems not to be aware of Olver’s facts."
Actually, I am aware of Olver's work and also Sören Illman's contribution.
Sören is an old friend and wrote to me somewhat miffed that I did not mention his work on the problem. What he proved was a very nice fact---that if a proper Lie group action is differentiable, then it can be made real analytic. As I pointed out in my article, there are very simple examples that Hilbert should have noticed (see my article---linked below---if you think I am being hard on Hilbert) that show that without properness this is false.
As for Olver, his contribution is also nice but a little off the beaten track. Here is a quick version. One facet of what Hilbert asked was whether a "local Lie group" (i.e., an open set in $R^n$ with a continuous group operation and inverse defined near the identity) could always be embedded in a global Lie group. Cartan answered that in a way that suffices for all practical purposes; he showed that after cutting back the original neighborhood to a smaller one it could be embedded. However a number of people (including Malcev and Douady) showed that without cutting back the answer could be "no". Their examples were infinite dimensional however, and Olver in his paper "Non-Associative Local Lie Groups" provided a class of finite dimensional examples.
OK, so why didn't I mention the work of Illman, Olver and a host of others who worked on the Fifth Proble after the 1950s. If you look at my article, available for download here:
the answer is clear. My article was part of a larger memorial article for Andy Gleason (my PhD advisor) and it was titled "Gleason's Contribution to the Solution of the Hilbert Fifth Problem". There was plenty to talk about there, and a discussion of other contributions to the Fifth Problem that happened decades later would have been out of place.
By the way, in regard to what is called "Route B" in an answer above, the first section of my article is titled "What IS Hilbert's Fifth Problem" in which I try to explain at least a little bit about how and why Hilbert's original statement of the Fifth Problem morphed over time.