The EU Center of Excellence for Exascale in Solid Earth (ChEESE) develops exascale transition capabilities in the domain of Solid Earth, an area of geophysics rich in computational challenges embracing different approaches to exascale (capability, capacity, and urgent computing). The first implementation phase of the project (ChEESE-1P; 2018-2022) addressed scientific and technical computational challenges in seismology, tsunami science, volcanology, and magnetohydrodynamics, in order to understand the phenomena, anticipate the impact of natural disasters, and contribute to risk management. The project initiated the optimisation of 10 community flagship codes for the upcoming exascale systems and implemented 12 Pilot Demonstrators that combine the flagship codes with dedicated workflows in order to address the underlying capability and capacity computational challenges. Pilot Demonstrators reaching more mature Technology Readiness Levels (TRLs) were further enabled in operational service environments on critical aspects of geohazards such as long-term and short-term probabilistic hazard assessment, urgent computing, and early warning and probabilistic forecasting. Partnership and service co-design with members of the project Industry and User Board (IUB) leveraged the uptake of results across multiple research institutions, academia, industry, and public governance bodies (e.g. civil protection agencies). This article summarises the implementation strategy and the results from ChEESE-1P, outlining also the underpinning concepts and the roadmap for the on-going second project implementation phase (ChEESE-2P; 2023-2026).