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The Milky Way's central molecular zone (CMZ) has emerged in recent years as a unique laboratory for the study of star formation. Here we use the simulations presented in Tress et al. 2020 to investigate star formation in the CMZ. These simulations resolve the structure of the interstellar medium at sub-parsec resolution while also including the large-scale flow in which the CMZ is embedded. Our main findings are as follows. (1) While most of the star formation happens in the CMZ ring at $R\gtrsim100 {\, \rm pc}$, a significant amount also occurs closer to SgrA* at $R \lesssim 10{\, \rm pc}$. (2) Most of the star formation in the CMZ happens downstream of the apocentres, consistent with the "pearls-on-a-string" scenario, and in contrast to the notion that an absolute evolutionary timeline of star formation is triggered by pericentre passage. (3) Within the timescale of our simulations ($\sim100$ Myr), the depletion time of the CMZ is constant within a factor of $\sim2$. This suggests that variations in the star formation rate are primarily driven by variations in the mass of the CMZ, caused for example by AGN feedback or externally-induced changes in the bar-driven inflow rate, and not by variations in the depletion time. (4) We study the trajectories of newly born stars in our simulations. We find several examples that have age and 3D velocity compatible with those of the Arches and Quintuplet clusters. Our simulations suggest that these prominent clusters originated near the collision sites where the bar-driven inflow accretes onto the CMZ, at symmetrical locations with respect to the Galactic centre, and that they have already decoupled from the gas in which they were born.