The Bullet Cluster (1E 0657-56), located at a redshift of 0.296, is among the best-known merging galaxy clusters and a key laboratory for dark matter studies. Although the mass distribution in the Bullet Cluster has been modelled using an increasing number of multiply imaged galaxies, only six systems had spectroscopic redshifts published prior to this work, which are essential for system confirmation and as lens model constraints. We present an updated gravitational lens model of the Bullet cluster, obtained by combining JWST NIRCam imaging and NIRSpec spectroscopy. Our lens model has been constrained on the basis of a catalogue of 135 secure multiple images from 27 background galaxies with spectroscopic redshifts, uniformly covering both subclusters and a wide redshift range of 0.9-6.7. We also provide a catalogue of 199 multiple image candidates. We modelled the cluster with the parametric lens modelling code Lenstool. We incorporated several large-scale halos, cluster member galaxies, intracluster gas, and group-scale halos surrounding the cluster core, motivated by spectroscopic studies of cluster member kinematics. We describe the main cluster component with a complex, elongated double-peaked distribution, along with the subcluster, modelled using a single large-scale halo coinciding closely with the brightest cluster galaxy at a projected separation of 4_-2_^+3^kpc. The uncertainty of the displacement has been improved three-fold thanks to the addition of JWST systems. The addition of group-scale substructures, roughly following the two axes of cluster assembly, improves the fit to the multiple image positions and provides a physically motivated alternative to a constant shear component. Our lens model shows the closest agreement with previous studies in aperture mass profiles at ~60kpc from the brightest cluster galaxies (BCGs), but exhibits significant differences in the detailed mass distribution as a result of different lens-modelling strategies and adopted constraints. The differences are reflected in small, but spatially coherent deviations between the new spectroscopic redshifts and redshifts predicted by earlier lens models.