This paper evaluates the impact of the fiducial cosmology assumed in the BAO analysis of the DESI DR1 data on the final measurements in DESI 2024 III. The study uses mock galaxy catalogues that mimic the DESI DR1 tracers (BGS, LRG, ELG, QSO) and spans a redshift range of 0.1 to 2.1. Four secondary ABACUSSUMMIT cosmologies are compared against DESI's fiducial cosmology (Planck 2018). These cosmologies include a lower cold dark matter density, a thawing dark energy universe, a higher number of effective species, and a lower amplitude of matter clustering. The mocks are processed through the BAO pipeline with consistent grid, template, and reconstruction cosmologies. The study determines a conservative systematic contribution of 0.1% to the error for both isotropic and anisotropic dilation parameters α_iso and α_AP. The results show that the fiducial cosmology has a minimal impact on the final BAO measurements, with the largest effect observed in the thawing-DE cosmology. The study also finds that the differences in the BAO scales are small and consistent across different cosmologies, with the main systematic error coming from the fiducial cosmology assumption. The results suggest that the BAO analysis is robust to the choice of fiducial cosmology, with the systematic error budget being dominated by the fiducial cosmology assumption. The study concludes that the BAO analysis is robust to the choice of fiducial cosmology, with the systematic error budget being dominated by the fiducial cosmology assumption.This paper evaluates the impact of the fiducial cosmology assumed in the BAO analysis of the DESI DR1 data on the final measurements in DESI 2024 III. The study uses mock galaxy catalogues that mimic the DESI DR1 tracers (BGS, LRG, ELG, QSO) and spans a redshift range of 0.1 to 2.1. Four secondary ABACUSSUMMIT cosmologies are compared against DESI's fiducial cosmology (Planck 2018). These cosmologies include a lower cold dark matter density, a thawing dark energy universe, a higher number of effective species, and a lower amplitude of matter clustering. The mocks are processed through the BAO pipeline with consistent grid, template, and reconstruction cosmologies. The study determines a conservative systematic contribution of 0.1% to the error for both isotropic and anisotropic dilation parameters α_iso and α_AP. The results show that the fiducial cosmology has a minimal impact on the final BAO measurements, with the largest effect observed in the thawing-DE cosmology. The study also finds that the differences in the BAO scales are small and consistent across different cosmologies, with the main systematic error coming from the fiducial cosmology assumption. The results suggest that the BAO analysis is robust to the choice of fiducial cosmology, with the systematic error budget being dominated by the fiducial cosmology assumption. The study concludes that the BAO analysis is robust to the choice of fiducial cosmology, with the systematic error budget being dominated by the fiducial cosmology assumption.