Spatial statistics often assumes that the spatial field of interest is stationary and its covariance has a simple parametric form, but these strong assumptions are not appropriate in many applications. We propose nonstationary and nonparametric Bayesian inference on the covariance matrix for a spatial field, with a prior distribution that shrinks toward popular Matern-type covariances. Our prior is motivated by recent results on the so-called screening effect for such covariances, which ensures exponential decay of the entries of the Cholesky factor of the precision matrix under a specific ordering scheme. This decay also results in approximate sparsity of the Cholesky factor, so that the number of nonzero entries to be estimated and the resulting computational complexity are both linear in the number of spatial locations. We apply our methodology to output from global climate models, enabling cheap statistical emulation of these computationally expensive physical models.