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We here present the results from a detailed analysis of nebular abundances of commonly observed ions in the collisional ring galaxy Cartwheel using the Very Large Telescope (VLT) Multi-Unit Spectroscopic Explorer (MUSE) dataset. The analysis includes 221 HII regions in the star-forming ring, in addition to 40 relatively fainter H$α$ emitting regions in the spokes, disk and the inner ring. The ionic abundances of He, N, O and Fe are obtained using the direct method (DM) for 9, 20, 20, and 17 ring HII regions, respectively, where the S$^{++}$ temperature-sensitive line is detected. For the rest of the regions, including all the nebulae between the inner and the outer ring, we obtained O abundances using the strong-line method (SLM). The ring regions have a median $12+\log\rm{\frac{O}{H}}$=8.19$\pm$0.15, $\log\rm{\frac{N}{O}}=-$1.57$\pm$0.09 and $\log\rm{\frac{Fe}{O}}=-$2.24$\pm$0.09 using the DM. Within the range of O abundances seen in the Cartwheel, the N/O and Fe/O values decrease proportionately with increasing O, suggesting local enrichment of O without corresponding enrichment of primary N and Fe. The O abundances of the disk HII regions obtained using the SLM show a well-defined radial gradient. The mean O abundance of the ring HII regions is lower by $\sim$0.1 dex as compared to the extrapolation of the radial gradient. The observed trends suggest the preservation of the pre-collisional abundance gradient, displacement of most of the processed elements to the ring, as predicted by the recent simulation by Renaud et al. (2018), and post-collisional infall of metal-poor gas in the ring.