The way the question was actually phrased:
"If black holes collided billions of years ago, I would assume that any gravity wave disturbance occurred at that time and has been winging its way through the universe ever since. We may be able to "visualize" this occurrence, but will it be taking those billions of years for any disturbance to reach us as well? Perhaps any crazy distortions have already occurred here and we don't know it; or may move slower than light speed and show up at some far distant time in the future."
This question is very well thought out. You're asking if gravity waves travel at the speed of light, or perhaps faster or slower. A few years ago, someone asked a similar question: "If the Sun were to vanish at once, would its gravity vanish instantaneously, or would we remain bound to the Sun's gravity for eight and a half minutes, the time that light requires to travel from the Sun to Earth?"
We are certain that gravity waves don't propagate instantaneously through the cosmos, as that would constitute a violation of Einstein's tenet that information cannot travel faster than light speed. So, if the Sun were to suddenly disappear from the physical Universe (cool idea!!), Earth would continue to orbit the Sun's region for at least 8.5 minutes, the minimum time required for the gravity waves to 'communicate' the Sun's absence to us. Similarly, the gravity waves from colliding black holes would require at least billions of years to reach us.
As gravity waves have so far gone undetected, physicists have been unable to directly determine their velocity.* If they did detect gravity waves at the same time as they observed the event producing them, such as a black hole collision, they would conclude that gravity waves propagate at light speed. Although nobody has yet "captured" a gravity wave, astronomers have other evidence to suggest that gravity waves obey light speed. For instance, they have observations of binary pulsars,** in orbit around each other. General Relativity predicts that their rapid orbits should produce gravity waves moving away at light speed. As a consequence of this wave production, the pulsars should be drawing closer to each other. Observations of binary pulsar PSR 1913+16 have confirmed that the pulsars are approaching one another in a manner predicted by GR if the gravity waves are, indeed, moving at light speed.
Many physicists would assert that gravity waves propagate as quickly as light. If the black holes believed to comprise Quasar PSO J334.2028+01.4075 do collide and produce detectable gravity waves that arrive at the same that astronomers observe the copious light the collision produces, we'll know that gravity waves travel at the velocity of light.
I hope this answer proves helpful.
*Detecting gravity waves has proven to be exceedingly difficult! Even waves produced by catastrophic cosmic events would be highly attenuated by the time they reached Earth. Detectors would have to be sensitive enough to detect an atomic-width disturbance in a laser beam more than a mile long.