This sort of scenario is quite possible, and would likely be the result of the merger of two supermassive black holes during the collision of the galaxies. We have evidence of this in the quasar 3C 186 (see Chiaberge et al. 2017). Over the course of about two billion years, two supermassive black holes circled around each other, emitting gravitational waves. The final burst, as they combined, was likely anisotropic, emitted in a particular direction. This propelled the resulting black hole the opposite way, ejecting it from the galaxy (although it's still nearby; it's only been about 5 million years since the merger).
I pick 3C 186 because we're fairly sure it's been ejected from its host. That's because it is spatially offset from the host galaxy's center by 10-11 kpc, and because it has a velocity offset, traveling towards us at about 2000 km/s, although its overall velocity vector does not point directly at us. Other candidates simply have only spatial or velocity offsets - not both.
If we use 3C 186 as a model, we have some parameters we can look at and analyze:
- Radial velocity: $\sim$2000 km/s
- Mass: $\sim10^9M_{\odot}$
- Emission: Mainly from the broad line region around the black hole
- Luminosity: $2.6\times10^{13}L_{\odot}$
What's notable is that the active galactic nucleus stayed active. The supermassive black hole was ejected along with high-velocity clouds orbiting close to it. That's why were able to still observe it, and compare its redshift with that of its former host galaxy. It's unclear how long this emission can continue, of course, but if the black hole approaches us fairly soon after it's ejected, we should still see emission from the broad line region and possibly from relativistic jets.
Let's say that it's been a long time since the black hole was ejected, and the gas and dust around it has long since been depleted. In this case, we have a compact object with the mass of a small dwarf galaxy headed our way. We should be able to observe it via gravitational microlensing. Since the angular size of an Einstein ring scales with the square root of the mass of the lens, we should observe lenses about $\sim10^4$ times larger than those created by stellar-mass black holes:
$$\theta_E=\sqrt{\frac{4GM}{c^2}\frac{d_O-d_L}{d_Od_L}}=\sqrt{\frac{4GM}{c^2}\left(\frac{1}{d_L}-\frac{1}{d_O}\right)}$$
where $d_O$ and $d_L$ are the distance to the lensed object and the distance to the lens, respectively. Say we observe the lens while the black hole is in intergalactic space - maybe between us and Andromeda. The lensed object, presumably a star in Andromeda, would have $d_O\approx780\text{ kpc}$. If we pick a resolution of $\theta_E\approx0.4$ arcseconds, then we find $d_L\approx768\text{ kpc}$. In other words, if the black hole was coming at us from Andromeda, we could see it from pretty far away!
That said, such an alignment would be unlikely. It's more probable that the supermassive black hole would be coming from another direction - say, from the Virgo Cluster, 18 Mpc away. This means we would see the black hole from 13.3 Mpc away at the most. In general, the distance to the lens at which the ring would have a radius of $\theta_E$ at the critical value is
$$d_L=\frac{d_O}{\frac{\theta_E^2c^2}{4GM}d_O+1}$$
and you can check my calculations for the given figures. It's even more likely that the black hole would not be in front of any source even mere tens of megaparsecs away. This of course would make it harder to detect, as the lensed object might appear dimmer, and the ring might be smaller.
The optimal direction for the black hole to sneak up on us from would be from a region of the sky we can't easily observe. I would recommend the Zone of Avoidance, where much of the sky is obscured by gas and dust in the Milky Way. This makes it very hard to perform observations of background galaxies, let alone detect lensing. We would likely need to see lensing from the IC 342/Maffei group, which lies about 3.3 Mpc away. Within 3 Mpc, the lensing would show up, but at that distance, the images would likely be blocked by the Zone of Avoidance.
I don't know how close it would be before we could make that detection; I'm not sure how to calculate it. I assume, though, that the distance would be greater than the distance at which the black hole would gravitationally affect the Milky Way (recall that its mass is comparable to a middling dwarf galaxy). I will work on calculating that range, if I can. But I suspect strongly that microlensing is the best detection method, and that the Zone of Avoidance is the optimal approach. I just need to determine how to combine extinction with lensing.
