I'm just curious. I've no physics eductation beyond GCSE. I've searched and searched for an answer but haven't been able to find one. I'm guessing it's at the speed of light??
To elaborate; I'm visualising the "rubber sheet" representation of spacetime, deformed to extreme by a black hole. As the black hole moves, the spacetime it moves into deforms accordingly - and the spacetime it just left returns to flat/undeformed. But how quickly does that return happen?
As others (and yourself) have said, disturbances in spacetime move at the speed of light. That includes the returning of spacetime to it's initial position. However, that's not the complete story, because in a blackhole, the masses are large enough to cause spacetime to jiggle. But then your question is something like "how long does it take for a pendulum to stop swinging?"
This question is not, in general, specific enough to answer. But in the famous LIGO chirp, the gravitational waves that emanated from the jiggling took about 0.1 seconds to settle down to the point where we could no longer detect them. This conflates multiple ideas such as the sensitivity of our instruments, the size of the colliding blackholes, and the distance between us and them.
This was an event that was somewhat extreme though. For a (very) brief moment it was brighter than all the stars in the universe.
So, to answer your question: in general it never returns but continues to jiggle forever. But in almost all circumstances and with the measurable precision that we have, it returns immediately (speed of light). Somewhat similar to the length of a piece of string
There are two velocities we are discussing here, one is the velocity of the transfer of information, the other the velocity of the change in space.
Let us take the example of an online movie of a satellite around the sun. The information arrives with the delay of the velocity of light. How fast the satellite is moving is found by seeing its position against the stars.
In the case of gravity, in the model you have of a moving ball deforming a membrane, the velocity of a moving deformation is another way of measuring/describing mathematically the velocity of the ball. The membrane becomes flat depending on the velocity of elasticity transfer, probably acoustic velocity on the membrane. Which is the analogue for the velocity of light in spacetime described by general relativity.
As the answer of Dr Xorile says, there will always be a deformation, because the masses of the bodies in space time are what create the deformation, but for large distances, which will depend on the size of the mass, it will no longer be possible to measure the deformation. In Newtonian gravity it means that the gravitational field of that mass is not measurable because it is too far away from the measuring point.
One specific example is a non-rotating black hole, which is described by Schwarzschild spacetime. Suppose the black hole formed from a collapsing star. Now intuitively, we often think of the star's mass as being contained at the centre of the black hole. However technically, this is a vacuum solution: there is no mass anywhere! $r=0$ is a singularity, various quantities are "1/0" there, it is not part of spacetime, and in particular we can't say if there is any mass there. In summary, we have a vacuum solution, but it is not springing back to flat spacetime.
[Disclaimer: what I have said is pretty standard, but it may be the research has moved on. Indeed, a quick search for use of the Dirac delta function in Schwarzschild spacetime turned up an interesting blog post on this.]
