The aim of the present study is to comprehensively analyse scale effects on the mechanical characteristics of carbon nanotubes (CNTs) with viscoelastic properties. A scale-dependent coupled longitudinal-transverse nonlinear formulation is presented for this aim. The roles of both the longitudinal and transverse motions as well as the viscosity effect due to the internal loss of the total energy are taken into account. The influence of large deformations due to the geometric nonlinearity is also taken into account. The nonlocal strain gradient theory (NSGT) is applied so as to describe scale effects on the mechanical characteristics of viscoelastic CNTs. Compared to the classical nonlocal theory, the NSGT better estimates scale effects since it is able to describe both the stiffness-hardening and -softening behaviours. The Kelvin-Voigt approach is used to capture the influence of the internal energy loss. Application of the NSGT together with the Hamilton principle yields the energy potential, the external work and the coupled longitudinal-transverse equations of the CNT. To determine an accurate numerical solution, the Galerkin scheme of discretisation and a continuation approach are finally utilised. The role of different parameters of the nanosystem in the nonlinear coupled mechanics of viscoelastic CNTs is discussed.