The solar photospheric abundance of oxygen is still a matter of debate. For about ten years some determinations have favoured a low oxygen abundance which is at variance with the value inferred by helioseismology. Among the oxygen abundance indicators, the forbidden line at 630nm has often been considered the most reliable even though it is blended with a NiI line. In Papers I and Paper II of this series we reported a discrepancy in the oxygen abundance derived from the 630nm and the subordinate [OI] line at 636nm in dwarf stars, including the Sun. Here we analyse several, in part new, solar observations of the the centre-to-limb variation of the spectral region including the blend at 630nm in order to separate the individual contributions of oxygen and nickel. We analyse intensity spectra observed at different limb angles in comparison with line formation computations performed on a CO5BOLD 3D hydrodynamical simulation of the solar atmosphere. The oxygen abundances obtained from the forbidden line at different limb angles are inconsistent if the commonly adopted nickel abundance of 6.25 is assumed in our local thermodynamic equilibrium computations. With a slightly lower nickel abundance, A(Ni)~6.1, we obtain consistent fits indicating an oxygen abundance of A(O)=8.73+/-0.05. At this value the discrepancy with the subordinate oxygen line remains. The derived value of the oxygen abundance supports the notion of a rather low oxygen abundance in the solar hotosphere. However, it is disconcerting that the forbidden oxygen lines at 630 and 636nm give noticeably different results, and that the nickel abundance derived here from the 630nm blend is lower than expected from other nickel lines.