Who's asking: Paul Guyot, St. Louis, MO
This question revealed a major gap in my circle of acquaintances: I don't know any sheep farmers. I was going to seek some out, but that will take a while, and I'm traveling too much this month.
So here's my usual combination of online research and making stuff up.
First off, some people say "three shakes" instead of "two shakes." Three shakes? That's just silly. Avoid these people, along with people who say "between you and I," "utilize," and "to be honest with you..."
No one seems to know where the phrase comes from, but people generally agree that lambs can shake their tails pretty fast, making two shakes almost as fast as one shake -- so perhaps the implication is that the listener needs to be a little patient, but not very. The tone of it is always vaguely apologetic; we use it to refer to ourselves, and you'd never hear anyone ask for something to be done in two shakes of a lamb's tail.
What's very cool and unexpected is that "shake" has become an official time measurement in nuclear science. Nuclear engineers and astrophysicists use "shake" to mean 10 nanoseconds. (A nanosecond is 10 to the power of negative nine; 10 nanoseconds is ten to the power of negative eight. Admiral Grace Hopper, whom I had the honor to meet on two occasions, used to hand out lengths of wire just under a foot long, which demonstrated the distance light travels in a nanosecond.)
As for how fast I can shake my tail -- well, never mind. Momentum can be a frightening phenomenon.