  We study the Single-Parton-Scattering (SPS) production of double quarkonia
(J/psi+J/psi, J/psi+Upsilon, and Upsilon+Upsilon) in pp and pp(bar) collisions
at the LHC and the Tevatron as measured by the CMS, ATLAS, LHCb, and D0
experiments in the Colour-Evaporation Model (CEM), based on the
quark-hadron-duality, including Next-to-Leading Order (NLO) QCD corrections up
to alpha_s^5. To do so, we also perform the first true NLO --up to alpha_s^4--
study of the p_T-differential cross section for single-quarkonium production.
This allows us to fix the non-perturbative CEM parameters at NLO accuracy in
the region where quarkonium-pair data are measured. Our results show that the
CEM at NLO in general significantly undershoots these experimental data and, in
view of the other existing SPS studies, confirm the need for Double Parton
Scattering (DPS) to account for the data. Our NLO study of single-quarkonium
production at mid and large p_T also confirms the difficulty of the approach to
account for the measured p_T spectra; this is reminiscent of the impossibility
to fit single-quarkonium data with the sole 3S18 NRQCD contribution from gluon
fragmentation. We stress that the discrepancy occurs in a kinematical region
where the new features of the improved CEM are not relevant.
