What is the actual black hole merger speed?
It varies per object, and throughout the merger (and doesn't neatly increase linearly).
An object with a lot mass can not come close to the speed of light, that would require almost infinite energy. Extremely tiny objects, particles, can come close to the speed of light but not actually reach it. Light can go the speed of light, only in a vacuum. Gravity always travels at the speed of light, unlike everything else which goes slower.
The first animation shows what happens when two large objects approach each other, miss, and then don't slingshot away from each other - they become trapped in an extreme elliptical orbit.
The second animation shows what happens when the orbits are relatively close, > ~0.01 parsecs - instead of an elliptical orbit they have a somewhat circular-like orbit, around the barycenter.
There are many different trajectories and resulting orbits, the binaries are also affected by: inflow, outflow, and other objects entering and leaving the area of their orbits. No one answer, no two are alike.
The paper "Observation of Gravitational Waves from a Binary Black Hole Merger" (Feb 11 2016), by B. P. Abbott et al. offers one study of one merger, they provide this illustration:
Top: Estimated gravitational-wave strain amplitude from GW150914 projected onto H1. This shows the full bandwidth of the waveforms, without the filtering used for Fig. 1. The inset images show numerical relativity models of the black hole horizons as the black holes coalesce. Bottom: The Keplerian effective black hole separation in units of Schwarzschild radii ($R_S=2GM/c^2$) and the effective relative velocity given by the post-Newtonian parameter $v/c=(GMπf/c^3)^{1/3}$, where $f$ is the gravitational-wave frequency calculated with numerical relativity and $M$ is the total mass (value from Table 1).
In-between those distances occurs a situation referred to as the final parsec problem.
Final parsec problem
When two galaxies collide, the supermassive black holes at their centers do not hit head-on, but would shoot past each other if some mechanism did not bring them together. The most important mechanism is dynamical friction, which brings the black holes to within a few parsecs of each other. At this distance, they form a bound, binary system. The binary system must lose orbital energy somehow, for the black holes to orbit more closely or merge.
Initially, the explanation is easy. The black holes transfer energy to gas and stars between them, ejecting matter at high speed via a gravitational slingshot and thereby losing energy. However, the volume of space subject to this effect shrinks as the orbits do, and when the black holes reach a separation of about one parsec, there is so little matter left between them that it would take billions of years to orbit closely enough to merge - more than the age of the universe. Gravitational waves can be a significant contributor, but not until the separation shrinks to a much smaller value, roughly 0.01–0.001 parsec.
Nonetheless, supermassive black holes appear to have merged, and what appears to be a pair in this intermediate range has been observed, in PKS 1302-102. The question of how this happens is the "final parsec problem".
A number of solutions to the final parsec problem have been proposed. Most involve the interaction of the massive binary with surrounding matter, either stars or gas, which can extract energy from the binary and cause it to shrink. For instance, if enough stars pass close by to the orbiting pair, their gravitational ejection can bring the two black holes together much quicker than would otherwise be the case.
See the "final parsec problem" link for the whole webpage.
"... the gravitational time dilation ..."
Means: In the black hole's timeframe time moves forward normally for both the black hole and the observer in their own frames. In each other's frames the outside observer see the black hole's time seemingly come to a standstill while the black hole observer sees the outside observer's time pass very rapidly.
