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The elemental ratios of carbon, nitrogen, and oxygen in the atmospheres of hot Jupiters may hold clues to their formation locations in the protostellar disc. In this work, we adopt gas phase chemical abundances of C, N and O from several locations in a disc chemical kinetics model as sources for the envelope of the hot Jupiter HD 209458b and evolve the planet's atmospheric composition using a 1D chemical kinetics model, treating both vertical mixing and photochemistry. We consider two atmospheric pressure-temperature profiles, one with and one without a thermal inversion. From each of the resulting 32 atmospheric composition profiles, we find that the molecules CH4, NH3, HCN, and C2H2 are more prominent in the atmospheres computed using a realistic non-inverted P-T profile in comparison to a prior equilibrium chemistry based work which used an analytical P-T profile. We also compute the synthetic transmission and emission spectra for these atmospheres and find that many spectral features vary with the location in the disc where the planet's envelope was accreted. By comparing with the species detected using the latest high-resolution ground-based observations, our model suggests HD 209458b could have accreted most of its gas between the CO2 and CH4 icelines with a super solar C/O ratio from its protostellar disc, which in turn directly inherited its chemical abundances from the protostellar cloud. Finally, we simulate observing the planet with the James Webb Space Telescope (JWST) and show that differences in spectral signatures of key species can be recognized. Our study demonstrates the enormous importance of JWST in providing new insights into hot Jupiter's formation environments.