Context. The ongoing Gaia-ESO Public Spectroscopic Survey is using FLAMES at the VLT to obtain high-quality medium-resolution Giraffe spectra for about 105 stars and high-resolution UVES spectra for about 5000 stars. With UVES, the Survey has already observed 1447 FGK-type stars. Aims: These UVES spectra are analyzed in parallel by several state-of-the-art methodologies. Our aim is to present how these analyses were implemented, to discuss their results, and to describe how a final recommended parameter scale is defined. We also discuss the precision (method-to-method dispersion) and accuracy (biases with respect to the reference values) of the final parameters. These results are part of the Gaia-ESO second internal release and will be part of its first public release of advanced data products. Methods: The final parameter scale is tied to the scale defined by the Gaia benchmark stars, a set of stars with fundamental atmospheric parameters. In addition, a set of open and globular clusters is used to evaluate the physical soundness of the results. Each of the implemented methodologies is judged against the benchmark stars to define weights in three different regions of the parameter space. The final recommended results are the weighted medians of those from the individual methods. Results: The recommended results successfully reproduce the atmospheric parameters of the benchmark stars and the expected Teff-log g relation of the calibrating clusters. Atmospheric parameters and abundances have been determined for 1301 FGK-type stars observed with UVES. The median of the method-to-method dispersion of the atmospheric parameters is 55 K for Teff, 0.13 dex for log g and 0.07 dex for [Fe/H]. Systematic biases are estimated to be between 50-100 K for Teff, 0.10-0.25 dex for log g and 0.05-0.10 dex for [Fe/H]. Abundances for 24 elements were derived: C, N, O, Na, Mg, Al, Si, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Y, Zr, Mo, Ba, Nd, and Eu. The typical method-to-method dispersion of the abundances varies between 0.10 and 0.20 dex. Conclusions: The Gaia-ESO sample of high-resolution spectra of FGK-type stars will be among the largest of its kind analyzed in a homogeneous way. The extensive list of elemental abundances derived in these stars will enable significant advances in the areas of stellar evolution and Milky Way formation and evolution. Based on observations made with the ESO/VLT, at Paranal Observatory, under program 188.B-3002 (The Gaia-ESO Public Spectroscopic Survey, PIs Gilmore and Randich). Appendices are available in electronic form at http://www.aanda.org