We detail our work on improving the performance and scalability of key numerical methods in the high-fidelity spectral element code Neko on accelerated exascale machines. Eifficient preconditioners are essential in incompressible fluid dynamics; however, the most eifficient method (with respect to convergence) might be challenging to implement with good performance on an accelerator. We present our development of a GPU-optimised preconditioner with task overlapping for the pressure-Poisson equation, improving the preconditioner's throughput (in TDoF/s) by close to 60%. The new preconditioner is explained in detail, together with detailed performance studies on accelerated Cray EX platforms, including strong scalability studies on LUMI and Frontier.