Tungsten sputtering rates and density profiles predicted using the edge plasma codes EDGE2D-EIRENE and DIVIMP are found to agree within a factor of 4 with measurements of neutral and singly-ionized W spectral line emission in the JET low-field side (LFS) divertor, and within a factor of 2 with SXR, VUV, and bolometric calculations of the W density in the main plasma. The edge plasma W predictions are extended to the core plasma using JINTRAC integrated core-edge modelling. Prompt redeposition of W is identified as the primary reason for the discrepancy between predicted and measured W emission in the divertor. The studied plasmas include attached divertor conditions in L-mode and type-I ELMy H-mode plasmas typical for JET. To more accurately reproduce the spectroscopically inferred W sputtering rates in EDGE2D-EIRENE, imposing the experimentally observed Be concentration of order 0.5% in the divertor is necessary. However, the W density in the main plasma is predicted to be insensitive to whether or not W is sputtered by Be at the divertor targets. Instead, the majority of the predicted core W originated in L-mode from sputtering due to fast D charge-exchange atoms at the W-coated tiles above the LFS divertor, and in H-mode due to D and W ions at the targets during ELMs.