I have been unable to find papers that make an estimate of the total amount of time required for the complete evolution from the LUA.
However, I can cite a couple of important papers which show that some of the more complicated steps can be achieved in timescales that are relatively short.
The 1994 paper, (A Pessimistic Estimate of the Time Required for an Eye to Evolve* looks at an area which has been commonly cited as one of the biggest challenges to Evolution - the evolution of the eye. (Ironically, Darwin himself was one of the first people to raise this issue, and one of the first to propose an answer.)
They theoretically estimate how many generations would be required for an eye to evolve - not from an LUA, but from a patch of skin that is light-sensitive.
Even with a consistently pessimistic approach the time required becomes amazingly short: only a few hundred thousand years.
Rates of Evolution on the Time Scale of Evolutionary Process looks at the amount of evolutionary progress we have seen compared to the amount that might be possible, and concludes it is much less than it has had time for.
The evolutionary process is dynamic but operates within relatively narrow morphological constraints compared to the time available for change.
That means, even though things change, they don't change as much as they could in the time available - which isn't that surprising.
In particular, the author looks at mammals, starting in the Cenozoic (aka the Age of Mammals, 65 million years ago), and concludes the variation found is four orders of magnitude less (i.e. one ten-thousandth) than the time-scale permitted. Evolution is not constrained by time as much as it is constrained by form.
Neither of the papers look at the entire evolution of animals and plants back from our common ancestor, but they illustrate on smaller time-scales, that biological complexity (like the eye) and biological diversity (like all of the mammals on the Earth) can evolve in a small fraction of the time that is available to them.
With this understanding, it becomes more comfortable to extrapolate and see that the 3.6 billion years is rather a long time, even in evolutionary terms.