To cope with the formidable sampling rate required by Nyquist criterion, compressive sensing (CS) has been recently adopted for ultrawideband (UWB) channel estimation. In this paper, exploiting the statistical sparsity of real UWB signals in the basis formed by eigenvectors, we develop a new CS dictionary called eigendictionary, which enables the use of CS for UWB channel estimation. With respect to the eigendictionary, the expansion vector of UWB signals is sparse and exhibits an additional structure in the form of statistically significant coefficients occurring in clusters. Capitalizing on this structure, we propose two novel Bayesian CS (BCS) algorithms to efficiently reconstruct UWB signals from a small collection of random projection measurements. Furthermore, by utilizing the common sparsity profile inherent in UWB signals, we extend the proposed Bayesian algorithms to multitask (MT) versions, which can simultaneously recover multiple UWB signals if available. Since the statistical connection between different UWB signals is exploited, the developed MT-BCS can obtain better performance than the single-task version. Extensive simulations using real UWB data show that the proposed schemes considerably reduce the requirement on sampling rate and present excellent performance compared with the traditional correlator and other CS-based channel estimation schemes.