We discuss the exclusive photoproduction of a heavy vector quarkonium, namely $J/\psi$ and $\Upsilon$ at 1-loop in $\alpha_s$. In collinear factorisation (CF), the amplitude for such a process is obtained by the convolution of a hard partonic sub-amplitude, with a universal generalised parton distribution (GPD). For the first time, we perform a complete calculation at 1-loop including full leading-log (LL) GPD evolution. We first demonstrate the huge instability of the cross section at high energies when the factorisation scale $\mu_F$ is varied. This instability has been reported previously in the literature, and occurs due to the large logarithms generated by the huge difference between the hard scale of the process, which is the mass of the heavy quarkonium here, and the centre-of-mass energy of the process. This problem was also reported in \textit{inclusive} heavy vector quarkonium photoproduction. We show that this issue can be resolved by resumming these large logarithms using high-energy factorisation (HEF), by performing a matching with the result in CF using the doubly logarithmic approximation (DLA) in order to be consistent with the fixed order evolution of GPDs. Finally, we show that the cross sections obtained from such a matching, besides being free from the previously mentioned factorisation scale variation instabilities, are consistent with the H1 data for $J/\psi$ production and with the ZEUS data for $\Upsilon$ production.