Nuclear-transition laser
There is now some overlap between what we can do using optical elements, and the sort of photon energies that correspond to nuclear excitations. In particular, as mentioned in e.g. this paper, the 229Th nucleus has a low-lying excited state at energy $E=7.8\pm0.5\:\mathrm{eV}$, and this is low enough to be accessible to some reasonably intense laser sources.
However, that already tells you where you stand with respect to making a laser at those same frequencies, and the very first thing you need to contend with is absorption. What we've found when performing optical experiments in the extreme-UV range is that light at those frequencies really doesn't like changing directions: it just wants to steamroll on ahead no matter what. Thus, you can make grazing-incidence optics with reasonably high reflectance but with incidence angles in the 85° range, or you can make normal-incidence mirrors with reflectances in the 10% range, but that's about the best you can do.
Neither of those options is particularly suited to building a resonant cavity (femtosecond enhancement cavities notwithstanding), so making a multipass laser is sort of ruled out, and it will only get worse as you increase the photon energy and more and more materials just give up and either absorb your photons or let them through unchanged.
The loss of a cavity is really quite a big loss, though it is not fatal. You can, in fact, make cavityless lasers, like e.g. modeless dye lasers, mostly by pumping hard enough that any spontaneously emitted photons will pick up a nontrivial amount of gain on its way out of the material, but this obviously makes it much harder to control the collimation and coherence of the radiation, which is presumably what you were doing this for in the first place.
Moreover, the "pumping hard enough" step is obviously extra challenging if your gain medium is excited nuclei. It is probably possible, if you work hard enough, to produce population inversion in a laser, by using e.g. an electron/neutron/alpha beam at exactly the right energy, but asking this step to be strong enough that the gain medium can undergo single-pass lasing is probably just too much.
However, if all you want is light at really high frequencies that is highly collimated and coherent, there are plenty of other options that you can look at.
- On the optical side, high-order harmonic generation can produce coherent photons all the way over to the $1\:\mathrm{keV}$ regime by combining something several thousand tiny photons into a single huge one through an extremely nonlinear interaction in a gas (reference).
- In a similar range you have free-electron lasers, which produce pretty coherent light by zigzagging high-energy electron beams with magnetic fields; these are easier to drive up to higher photon energies and pulse energies but they are sometimes a bit jittery on the pulse timing. If you're wondering whether this is indeed a laser or not, we've got
an appa question on that. - As another bit of really impressive technology, you can also have plasma-based soft-x-ray lasers (also here), for which you basically use a huge laser to ionize some plasma to really high charge states, create a population inversion in some electronic transition in the ion (which will be in the keV regime or more, since you've essentially stripped away a full shell or more), and then use that to lase or to amplify some coherent seed that you got from somewhere else (as in e.g. this heroic experiment).
Given that you have all of those options available and working, if you want to develop a source based on a nuclear transition, you really need to ask yourself why it is you're actually doing it. At this point, really, the only advantage you could get by switching your gain medium to a nuclear transition is to extend the frequency range all the way over to a MeV and beyond, but then you also need to provide some actual use for such a source. There you're really stuck to nuclear physics experiments, because for every other kind of physics that kind of photon will either pass by, or destroy your system. Nuclear physics, however, seems to be doing just fine without that kind of source, and you'd be competing with beams of electrons / neutrons / protons / alphas / whatever, which have lots of available (and installed) technology and expertise already. Thus, it'd be a lot of investment for very uncertain gains, so you can see that the case for that research is pretty flimsy at the moment.
That said, if what you really want to know is whether stimulated emission is a thing you could actually observe in nuclear transitions, the term to look for is induced gamma emission, which has been considered as a possibility but which has yet to be conclusively demonstrated experimentally. It appears the closest we have gotten to this is via the decay of a metastable state of hafnium1: here, as I would understand it, you would produce 178Hf in some nuclear reactor, and some nontrivial fraction of it comes out in the metastable state 178m2Hf without the need for pumping. This state will normally decay with a 30-year half-life, so you've got some time to e.g. try to bombard the ground-state 178Hf with stuff that will transmute into something that can be chemically separated, giving you a sample enriched in the metastable isomer, which would then be susceptible to induced emission. However, it appears that for the moment this remains an unrealized possibility.
$^1$ hat-tip to @rob for pointing me in this direction; see the ensuing chat conversation for more details.