Oxygen abundances have been derived in a sample of very metal-poor stars from observations of the Oi triplet at /λλ 7771-5 Å and OH lines in the near UV. Iron abundances derived in LTE were corrected for NLTE effects following Thévenin and Idiart [ApJ 521 (1999) 753]. Furthermore, a detailed NLTE analysis of iron lines has been carried out for one of the observed stars, BD +23/° 3130 ([Fe/H]NLTE=-2.43), providing consistent values of effective temperature, surface gravity and metallicity, that are in good agreement with independent estimates from the infrared flux method, Hipparcos parallaxes and recent NLTE work in the literature, respectively. These parameters, especially the higher gravity obtained with respect to previous analyses, reduce the discrepancies claimed by Fulbright and Kraft [AJ 118 (1999) 527] between the oxygen abundances determined from OH and [Oi] /λ 6300 Å lines, and give a similar abundance from the Oi triplet for BD+23/° 3130. A mean value of the oxygen-to-iron ratio [O/Fe]/=0.78+/-0.16 for this star is fully consistent with the abundances derived from the three sets of features. This consistency, which is found using 1D hydrostatic model atmospheres, strongly constrains the effects predicted by 3D hydrodynamical models on these three indicators. The oxygen abundances derived for this new sample confirm previous findings for a progressive linear increase in the oxygen-to-iron ratio with a slope /-0.33+/-0.02 (including NLTE corrections to the iron abundances for all the stars considered) from solar metallicity to [Fe/H]~-3, and [O/Fe] values as high as ~1.1 for stars with [Fe/H]<-2.5. These results can be interpreted as evidence for oxygen overproduction in the very early epoch of the formation of the Galactic halo, possibly associated with supernova events with very massive progenitor stars.