If U238 ions were circulating in a particle accelerator would their neutron absorption cross section spectra change?
The neutron cross-section is a function of energy in the CoM frame for the pair: $$ \sigma = \sigma(E) ,$$ so unless the neutrons are boosted to match, they will be a different relative energy, and yes the cross-section changes.
But you haven't learned anything interesting by doing that.
If the question concerns neutrons that have been boosted to move more-or-less in coincidence with the Uranium, then the principle of relativity tells us that no the cross-section will not differ from that measured at rest in the lab frame.
Again, you haven't learned much.
what clearly matters is the relative speed of the ions and the neutrons that may be absorbed; you won't change anything fundamental about the reaction if both particles are moving by the same speed, by the principle of relativity. For reasonably small energies, this is usually parameterized by the relative velocity $v$. See a graph how the cross section depends on $v$:
Quite generally, the cross section decreases with $v$. For neutron kinetic energies below 1 eV (in the nucleus' rest frame), the cross section decreases as $1/v$ - it's the $1/v$ region. Then you have resonance peaks for neutron energy between $1$ and $100$ eV, before the cross section returns to a decrease that is getting slower and slower.
The graph is only drawn up to neutron energies of order $1$ MeV. You don't really want to go to GeVs or even TeVs because the kinetic energy will then be large enough to create pions and the process itself will be dominated by the usual "collider-like" inelastic collisions rather than a simple absorption which is only an important process for relatively small relative velocities.
Of course that if you accelerate neutrons against ions so that the center-of-mass energy will be in hundreds of GeVs or TeVs, then you will have a standard hadron collider and the results won't be too different from those at the LHC. After all, protons and neutrons don't differ too much. Only a tiny fraction of events will look like a smooth "nuclear physics" absorption. At very high energies and relativistic speeds, it is simply inappropriate to think about the processes in terms of nuclear physics - neutrons and ions - which is just an effective theory for small energies (and velocities). You should think in terms of subnuclear i.e. particle physics - it's a collision of high-energy quarks and gluons, just like at the LHC. That's why the science about quarks etc. belongs to something called the high-energy physics: you need it when the energies are high!
Nuclear physics - such as neutron absorption - is pretty much a low-energy physics even though it is 1 million times higher an energy than the truly low-energy physics which contains atomic physics, chemistry, and biology. The low-energy physics concepts are only good enough approximations to describe the world when the energies are low enough.
Best wishes Lubos