I would be quite surprised if a Nicoll-Dyson beam is capable of moving a stellar-mass or supermassive black hole without taking many centuries (or millions of years) to do so.
Let's say our black hole has a mass 10 million times that of the Sun, and let's say we want it to reach a velocity of $\sim100\;\text{km/s}$ relative to its current rest frame, which I'd say is a reasonable velocity by the standards of astronomical objects. If our beam was to capture and reuse all of the output power of the Sun, it would need to collect solar energy for a time
$$\tau=\frac{\frac{1}{2}Mv^2}{P}=\frac{\frac{1}{2}10^7M_{\odot}(100\;\text{km/s})^2}{L_{\odot}}\approx10^{13}\;\text{years}$$
If we choose a more luminous star, we might be able to reduce that by maybe 4-5 orders of magnitude, but it still involves gathering energy for 100 million to 1 billion years. If we choose an ensemble of stars, maybe we can force that down by a couple of orders of magnitude, but it's still rather high.
This becomes drastically more manageable in the case of a stellar-mass black hole, which might weigh in at a few tens of solar masses. Now we're looking at gathering energy from a luminous star for only a few centuries. However, we have a new problem: It's quite hard to focus all of that energy onto a stellar-mass black hole. A black hole of $M=10M_{\odot}$ is about 60 km in diameter. Taking into account the fact that its gravity will drastically bend spacetime, I'd argue that its true cross-section is a bit larger, but not by much.
This means that only a fraction of our beam will actually transfer its energy to the black hole, and it will in turn take much more time than expected to help it reach the desired speed. Even if our beam is highly collimated, it will still spread out enough over interstellar distances. I think we can now increase our wait time by at least an order of magnitude, likely more.
For lighter black holes, this may work. I believe a black hole of $\sim10^9$ kg will take less than a day to evaporate, so we can call that "stable" (and as the evaporation time scales as $t\propto M^3$, let's assume that our tiny black hole is no less massive than that by a factor of a few). Now, if we were to focus all the light of the Sun on that black hole, we could reach the desired speed in about a nanosecond. Indeed, to get it to more human-sized speeds, we could perhaps even do without a source anywhere near as luminous as the Sun.
It's tempting to explore other options which wouldn't work for the larger, more massive astrophysical black holes, like particle accelerators. If the black hole was charged, we could use magnetic fields to accelerate it to high speeds. Unfortunately, this would require a substantial amount of energy. Our black hole is $10^{34}$ times as massive as a proton, and our particle accelerators are presumably incapable of accelerating an object that massive to any significant speed.
As a concrete example, a cyclotron must output an energy of
$$E=\frac{q^2B^2R^2}{2M}$$
to move a particle of charge $q$ and mass $M$ in a circle of radius $R$ in a magnetic field $B$. We see that we then have to make up 34 orders of magnitude in the numerator. Even if the black hole was significantly charged, we would need an enormous accelerator with strong magnetic fields, which I find unlikely.
Now, you want our black holes to be larger than atoms. If you're talking about that in terms of mass, well, we're already at the lower limit of black holes which can remain stable for significant periods of time. If you're talking about that in terms of radius, we'd need a black hole with a Schwarzschild radius of about an ångström, or $10^{-10}$ meters. This requires a black hole of mass greater than $M>6.7\times10^{16}\;\text{kg}$, which poses even more problems than before. I think this definitively rules out particle accelerators, and all reasonable methods using present-day technology for moving tiny things very quickly.
