Discretizing a Fredholm integral equation of the second kind with kernel k using collocation or Galerkin's method leads to a linear system whose size depends on the dimension of the chosen trial space, but not on the quadrature rule used to integrate over a bounded multivariate domain Omega in R^d. This contrasts with the classical Nyström method, where the degrees of freedom in the numerical solution are identified with the set of nodes Y of a global quadrature formula over Omega. This proves inefficient if the kernel k varies much more rapidly than the solution u: in such cases, collocation and Galerkin's methods can profitably combine a coarse trial space with arbitrarily fine quadrature, whereas the classical Nyström method must solve an unnecessarily large squared system with |Y| unknowns. In this talk, we introduce a high-order decoupled variant of the Nyström method, where the node set X for approximating u and the node set Y for discretizing the integral operator can be chosen independently. The case |X| << |Y| is of practical interest, because the decoupled scheme produces a squared system with only |X| unknowns. We prove, using standard tools from functional analysis, that arbitrarily high orders of convergence can be achieved under natural assumptions. Theoretical arguments and numerical experiments illustrate the computational advantages of the decoupled scheme, especially for rapidly-varying kernels. This work focuses on smooth kernels and 2D domains, paving the way for further generalizations. Numerical experiments are performed in a meshless setting, with X and Y being sets of scattered nodes over the closure of Omega. The nodes are generated by the advancing front method provided by the library NodeGenLib. Quadrature employs the recently developed high-order moment-free approach described in the doctoral dissertation of Bruno Degli Esposti. Applications to nonlocal population dynamics subject to logistic growth are presented.