Bottom-up Mergesort vs Natural Merge Sort
Natural Mergesort usually refers to Timsort. In terms of comparison to bottom-up mergesort, Timsort has $O(n)$ best-case running time where as bottom-up mergesort is at least $O(n\log n)$ on any input.
However, analysis of the worst-case running time of Timsort proves to be more tricky; an 2018 paper [1] establishes that its worst-case time complexity is $O(n\log \rho)$, where $\rho$ is the number of maximal monotonic sequences in the input array.
[1]: Nicolas Auger, Vincent Jugé, Cyril Nicaud, Carine Pivoteau, &quot;On the Worst-Case Complexity of TimSort&quot;