The self-organizing properties of nematic liquid crystals (LCs) can be used to align carbon nanotubes (CNTs) dispersed in them. In the previous paper [P. van der Schoot, V. Popa-Nita, and S. Kralj, J. Phys. Chem. B 112, 4512 (2008)], we have considered the weak anchoring limit of the nematic LC molecules at the nanotube's surface, where the CNT alignment is caused by the anisotropic interfacial tension of the nanotubes in the nematic host fluid. In this paper, we present the theoretical results obtained for strong enough anchoring at the CNT-LC interface for which the nematic ordering around nanotube is apparently distorted. Consequently, relatively strong long-range and anisotropic interactions can emerge within the system. In order to get insight into the impact of LC ordering on the alignment of nanotubes we treat the two mixture components on the same footing and combine Landau-de Gennes free energy for the thermotropic ordering of the liquid crystal and Doi free energy for lyotropic nematic ordering of carbon nanotubes caused by their mutually excluded volume. The phase ordering of the binary mixture is analyzed as a function of the volume fraction of the carbon nanotubes, the strength of coupling, and the temperature. We find that the degree of ordering of the nanorods can be tuned by raising or lowering the temperature or by increasing or decreasing their concentration.