We perform the first complete one-loop study of exclusive photoproduction of vector quarkonia off protons in Collinear Factorisation (CF) including the scale evolution of the Generalised Parton Distributions (GPDs). We confirm the perturbative instability of the cross section at high photon-proton-collision energies ($W_{\gamma p}$) at Next-to-Leading Order (NLO) in ${\alpha }_s$ and solve this issue by resumming higher-order QCD corrections, which are enhanced by a logarithm of the parton energies, using High-Energy Factorisation (HEF) in the Doubly-Logarithmic Approximation (DLA) matched to CF. Our NLO CF ⊕ DLA HEF results are in agreement with the latest HERA data and show a smaller sensitivity to the factorisation and renormalisation scales compared to Born-order results. Quark-induced channels via interference with gluon ones are found to contribute at most 20% of the cross section for $W_{\gamma p}>100$ GeV. Our results also show that such exclusive cross sections cannot be accurately obtained from the square of usual Parton Distribution Functions (PDFs) and clearly illustrate the importance of quarkonium exclusive photoproduction to advance our understanding of the 3D content of the nucleon in terms of gluons. Our work provides an important step towards a correct interpretation of present and future experimental data collected at HERA, the EIC, the LHC and future experiments.