Charge transfer cross sections and rate coefficients for ion-hydrogen collisions are fundamental input data for kilonova radiative transfer modeling, fusion edge plasma diagnostics, and heavy-ion accelerator beam lifetime evaluation. Despite this broad demand, systematic theoretical data for the U^q+ + H system at low energies were almost entirely absent before this work. In this study, total charge transfer cross sections and rate coefficients for single-electron capture in U^q+ (q = 3–92) + H(1s) collisions are investigated within the multichannel Landau-Zener (MCLZ) framework over the collision-energy range from 0.001 to 1000 eV/u, which represents the first unified treatment of nearly the entire uranium ionization sequence. The entrance-channel potential incorporates polarization attraction and short-range repulsion, while exit-channel potentials are constructed from Coulomb repulsion with asymptotic energy shifts derived from NIST spectroscopic data. Nonadiabatic coupling matrix elements are evaluated using the Butler-Dalgarno model for q \leqslant 4 and the Taulbjerg model for q \gt 4. Multichannel transition probabilities are computed via the recursive formula, and rate coefficients are obtained through Maxwell-Boltzmann thermal averaging over the temperature range 1–10⁵ K. For low charge states (q = 3–12), the cross sections approximately follow an E^-1 scaling below their charac-teristic turnover energies E_t and exhibit non-monotonic energy dependences at higher energies. The turnover energy generally shifts upward with increasing q. For q \gtrsim 13, no resolvable turnover is observed within the present energy range, and the cross sections approximately retain the E^-1 scaling behavior over the whole calculated range. The charge state dependence of the cross sections reveals pronounced shell structure modulation. Systematic minima occur at closed-shell (\mathrmd^10 and \mathrmf^14) and half filled shell (\mathrmf^7) configurations, as exemplified by q = 15, 33, 40, and 65. This behavior is attributed to exchange energy stabilization at these configurations, which produces anomalously high ionization energies, increases the energy defect, shifts the avoided crossing to smaller internuclear distances, and suppresses the electron-capture probability. The thermally averaged rate coefficients preserve these shell-structure oscillations, and those for high charge states decrease approximately as T^-1/2. The present dataset covers temperature ranges relevant to kilonova ejecta (10³–10⁴ K), fusion edge plasmas (10⁴–10⁵ K), and accelerator beam pipes from room temperature to 10³ K. It therefore provides systematic theoretical reference data for plasma modeling and beam-transport design. The dataset presented in this paper is openly available at https://doi.org/10.57760/sciencedb.j00213.00235.