We present a new determination of the evolving galaxy UV luminosity function (LF) over the redshift range \(8.5<z<15.5\) using a combination of several major Cycle-1 JWST imaging programmes - PRIMER, JADES and NGDEEP. This multi-field approach yields a total of \(\simeq370\) sq. arcmin of JWST/NIRCam imaging, reaching (5-\(\sigma\)) depths of \(\simeq30\) AB mag in the deepest regions. We select a sample of 2548 galaxies with a significant probability of lying at high redshift (\(p(z>8.5)>0.05\)) to undertake a statistical calculation of the UV LF. Our new measurements span \(\simeq4\) magnitudes in UV luminosity at \(z=9-12.5\), placing new constraints on both the shape and evolution of the LF at early times. Our measurements yield a new estimate of the early evolution of cosmic star-formation rate density (\(\rho_{\rm{SFR}}\)) confirming the gradual decline deduced from early JWST studies, at least out to \(z \simeq 12\). Finally we show that the observed early evolution of the galaxy UV LF (and \(\rho_{\rm{SFR}}\)) can be reproduced in a \({\rm \Lambda}\)CDM Universe, with no change in dust properties or star-formation efficiency required out to \(z \simeq 12\). Instead, a progressive trend towards younger stellar population ages can reproduce the observations, and the typical ages required at \(z \simeq\) 8, 9, 10, and 11 all converge on \(\simeq 380-330\) Myr after the Big Bang, indicative of a rapid emergence of early galaxies at \(z \simeq 12 - 13\). This is consistent with the first indications of a steeper drop-off in \(\rho_{\rm{SFR}}\) we find beyond \(z \simeq 13\), possibly reflecting the rapid evolution of the halo mass function at earlier times.