This article deals with the investigation of the effect of carbon nanotube (CNT) waviness on the effective thermal conductivities of a novel fuzzy fiber-reinforced composite (FFRC). The distinctive feature of the construction of this novel FFRC is that wavy CNTs are radially grown on the circumferential surfaces of the carbon fibers. Effective thermal conductivities of the FFRC are determined by developing the method of cells (MOCs) approach in conjunction with the effective medium (EM) approach. The effect of CNT waviness is studied when wavy CNTs are coplanar with either of the two mutually orthogonal planes of the carbon fiber. The present study reveals that (i) if CNT waviness is parallel to the carbon fiber axis then the axial ($K1$) and the transverse ($K2$) thermal conductivities of the FFRC are improved by 86% and 640%, respectively, over those of the base composite when the CNT volume faction present in the FFRC is 16.5% and the temperature is 400 K, (ii) the effective value of $K1$ of the FFRC containing wavy CNTs being coplanar with the carbon fiber axis is enhanced by 75% over that of containing straight CNTs for the fixed CNT volume faction when the temperature is 400 K, and (iii) the CNT/polymer matrix interfacial thermal resistance does not affect the effective thermal conductivities of the FFRC. The present work also reveals that for a particular value of the CNT volume fraction, optimum values of the CNT waviness parameters, such as the amplitude and the wave frequency of the CNT for improving the effective thermal conductivities of the FFRC can be estimated.