This paper extends the worst-case norm (WCN) of linear systems subject to inputs with magnitude and rate bounds to the WCN of uncertain linear systems under the same inputs. While the WCN for linear systems can be accurately approximated by simply solving a sparse linear programming, the computation of the WCN for uncertain linear systems leads to an NP-hard problem. In this paper, a branch-and-bound algorithm is applied to calculate the WCN in the presence of uncertainty. Subsequently, we derive the bounds for two approximation errors, namely, the truncation error and the discretization error, which are resulted from the proposed WCN computation method. Based on these error bounds, we give a brief guideline for choosing appropriate values of the terminal time and the sampling time. Numerical examples demonstate that computation time of the proposed algorithm is reasonable within certain problem dimensions. An exhaustive search is employed to validate the branch-and-bound algorithm. Finally, we suggest a means to improve the WCN computation for problems with higher dimension.