We report the results of a non-LTE Fe, O and Mg abundance analysis of the carbon-nitrogen-rich ultra-metal-poor giants CS 29498-043 and CS 22949-037. The abundance of oxygen has been derived from measurements of the oxygen triplet at 7771-5 Å in high resolution spectra obtained with KeckI/HIRES and the forbidden line [O I] 6300 Å detected in the TNG/SARG spectra of CS 29498-043. Detailed non-LTE analysis of Fe lines has provided reliable stellar parameters which, however, do not resolve the oxygen abundance conflict as derived from the O I 7771-5 Å triplet and the [O I] 6300 Å forbidden lines. We obtained the following oxygen abundance: for CS 22949-037 [O/Fe] = 3.13, 1.95; and for CS 29498-043; [O/Fe] = 3.02, 2.49, based on the O I 7771-5 Å triplet and the [O I] 6300 Å forbidden line, respectively. A similar conflict appears to exist between the forbidden resonance line Mg C I 4571 Å and several subordinate lines, such as Mg I 5172 and 5183 Å. Our analysis demonstrates the failure of standard plane-parallel atmosphere models to describe the physical conditions in the line-forming regions of these ultra-metal-poor giants.