I've been mulling over the potential logic of the universe of Doctor Who for a while now, and I think it makes a lot more sense than it seems to at first glance. I mean, it's still wildly implausible. But I think some of the major categories of complaints might not be totally valid. My theory relies on the overlapping of the history of the universe many, many times -- which I think is fair, because it's a show about time travel. If we assume that the state of the universe as it exists is the most recent combination of all the different instances of time travel, the Doctor Who universe makes a lot of sense.
Part 1 of the theory: Time Lords are the many-universe-lengths descendants of humanity.
I think this one is pretty straight-forward. Humans tinker with time travel enough that they manage it on a scale to save the species, within the timeframe of a single stretch from existence to oblivion. Then, the Time Lords are the descendants of humans who most obsessively continued to tinker with time travel.
2. Most of the species throughout the universe are descendants of:
(a.) other strains of the early human time travelers, or
(b.) other earth-native species who were transported during the terraforming of other planets.
This would explain why so many species have so much in common with humanity -- anthropomorphism, speaking language, similar vulnerabilities and dietary needs, sometimes even the ability to interbreed. (The Daleks might be one of the only other real species in the universe. Or, they might just be some of the most horrifically corrupted humans.)
3. The Weeping Angels are the flew-too-close-to-the-sun poetically punished descendants of the most extremist Time Lord experimenters.
They have, as a fact of their existence, unlocked some of the most basic problems of the universe -- applying quantum indeterminacy to a human-scale survival technique, a biological ability to transport people in time, and they're said to be older than time itself -- which could easily be because they were created when some already too-far-gone Time Lords tried to travel outside of the existence of time.
I'm not sure if any of these theories have been articulated before, or who I should credit or nod to if they have. And they still don't explain the problems of some of the essentially impossible things that go on in Doctor Who -- though that can basically all be chalked up to "Time Lords are way more advanced than we can even begin to imagine."
I also like the vision of the timeline of the universe that this theory provides, because it seems to reflect the way it plays out in the show -- that there's a sort of horizontal flow of time, iterations of the universe, then there's a vertical flow of time that becomes increasingly chaotic and tenuous.
This theory also arguably self-explains why it's not explained in the show -- time is constantly being rewritten. Many of the processes of point A to point C could continue to exist long after point B has been overwritten a hundred times. Certain things remain constant, the strongest pillars of truth through time, and certain things become fixed.
As for "Wibbly-wobbly, timey-wimey," I imagine this all being far more complicated than I've just explained, well beyond my own understanding. Also: if I ever get to write an episode of Doctor Who, there will be a scene when he spends some time examining a holographic representation of time, trying to figure it out. It will be made clear that "Wibbly-wobly, timey-wimey" is the Doctor's way of saying "I failed that class, stop asking me questions about it."