We study the uniform convergence rate of the nonparametric maximum likelihood estimator (MLE) for the sub-distribution functions of failure times in the current status data with competing risks model. It is known that the MLE have (Formula presented.) -norm convergence rate (Formula presented.) in the absolutely continuous case, but no agreement has emerged on a uniform convergence rate. We specify conditions under which (Formula presented.) is the uniform convergence rate for the MLE of the sub-distribution functions on finite intervals. The obtained result refines known uniform convergence rate in the particular case of current status data. The main result is applied in order to get the uniform convergence rate of the MLE for the survival function of failure time in the current status right-censored data model.