Coalescent theory - independence of coalescent times
As long as members of a generation &quot;randomly choose&quot; their ancestor in the previous generation the law of independent probability (your equation) will hold.
Any study of coalescent theory begins with the Wright-Fisher model. The assumptions are:

finite diploid population of constant size N,
non-overlapping generations (simultaneous reproduction),
random mating,
no mutation, selection, or migration.

These assumptions are consistent with independent non-identically distributed waiting times. An example of an assumption under which independence no longer holds:
Individual B's random choice in generation 2 of ancestor A in generation 1 reduces the probability that individual C in generation 2 will choose A. In other words, the likelihood that A will give its genes to the next generation falls off with each new recipient. Then the independence no longer holds.
See, e.g., Deonier, Computational Genome Analysis (2005, Springer) at pp. 392 ff.
J. Wakely's paper Coalescent Theory: An Introduction (Systematic Biology, 58:1, Feb. 2009) may be one of the best overviews of this immense topic available. He mentions Kingman's 1982 mathematical proof (which I haven't gone through) of the coalescent process (Stochastic Processes and their Applications 13 (1982)--available as a free download from ScienceDirect).