We present deep echelle spectrophotometry of the Galactic HII regions M16, M20 and NGC 3603. The data have been taken with the Very Large Telescope Ultraviolet-Visual Echelle Spectrograph in the 3100-10400 Å range. We have detected more than 200 emission lines in each region. Physical conditions have been derived using different continuum and line intensity ratios. We have derived He+, C++ and O++ abundances from pure recombination lines as well as collisionally excited lines (CELs) for a large number of ions of different elements. We have obtained consistent estimations of the temperature fluctuation parameter, t2, using different methods. We also report the detection of deuterium Balmer lines up to Dδ (M16) and to Dγ (M20) in the blue wings of the hydrogen lines, which excitation mechanism seems to be continuum fluorescence. The temperature fluctuation paradigm agrees with the results obtained from optical CELs, and the more uncertain ones from far-infrared fine-structure CELs in NGC 3603, although, more observations covering the same volume of the nebula are necessary to obtain solid conclusions.