The invariant yields for \(J/\psi\) production at forward rapidity \((1.2<|y|<2.2)\) in U\(+\)U collisions at \(\sqrt{s_{_{NN}}}\)=193 GeV have been measured as a function of collision centrality. The invariant yields and nuclear-modification factor \(R_{AA}\) are presented and compared with those from Au\(+\)Au collisions in the same rapidity range. Additionally, the direct ratio of the invariant yields from U\(+\)U and Au\(+\)Au collisions within the same centrality class is presented, and used to investigate the role of \(c\bar{c}\) coalescence. Two different parameterizations of the deformed Woods-Saxon distribution were used in Glauber calculations to determine the values of the number of nucleon-nucleon collisions in each centrality class, \(N_{\rm coll}\), and these were found to give significantly different \(N_{\rm coll}\) values. Results using \(N_{\rm coll}\) values from both deformed Woods-Saxon distributions are presented. The measured ratios show that the \(J/\psi\) suppression, relative to binary collision scaling, is similar in U\(+\)U and Au\(+\)Au for peripheral and midcentral collisions, but that \(J/\psi\) show less suppression for the most central U\(+\)U collisions. The results are consistent with a picture in which, for central collisions, increase in the \(J/\psi\) yield due to \(c\bar{c}\) coalescence becomes more important than the decrease in yield due to increased energy density. For midcentral collisions, the conclusions about the balance between \(c\bar{c}\) coalescence and suppression depend on which deformed Woods-Saxon distribution is used to determine \(N_{\rm coll}\).