In general, heat transfer with internal flows - like flows in a pipe - does depend on Reynolds Number. For small enough Reynolds number, the flow is laminar and the nature of the laminar flow is such that fluid inertia has negligible influence, resulting in a constant value for the laminar Nusselt number, but only when the internal flow is "fully developed." For laminar flow in the entrance region of the pipe, the Nusselt number is not constant, and its value is different for different heat transfer configurations. This is explained more fully below.
The Reynolds number is a ratio of inertial forces to viscous forces, VD/Nu, where V is fluid velocity, D pipe diameter, and Nu fluid kinematic viscosity. When inertial forces are small enough, viscosity dominates, and the transfer of momentum across minute streamlines is solely because of friction (viscosity), and it occurs on a molecular level. At that level, the momentum is "diffused" across the streamlines, being transferred by molecule bumping into molecule. With larger Reynolds numbers, inertia becomes important and momentum transfer occurs at a larger length scale, on the order of small packets of fluid. Such packets, however small, still contain many billions of molecules.
With laminar flow, heat is also diffused by molecule bumping into molecule. The <PERSON> number is the ratio of convective heat transfer to conductive heat transfer, given by hD/k, where h is the film coefficient, D pipe diameter, and k fluid thermal conductivity. The film coefficient is a convenient definition, made in the simplest way to capture the essential physics; h = Q/(Ts - Tref), where Q is the bulk heat transferred, Ts the temperature of the heated/cooled surface, and Tref is a reference temperature, chosen for convenience. 362
Q is the heat transferred between the surface across the fluid to where the reference temperature is defined.
In the limit of laminar flow, there is really no convection, all the heat transfer is by conduction (diffusion), but not like conduction through a solid, because of the moving fluid. The motion produces a boundary layer and temperature profile, as well as a velocity profile. The Nusselt number is constant only when this temperature profile becomes fixed (fully developed). For the entrance region of the pipe, the profile is still developing, and the laminar Nusselt number is not constant, changing with axial distance.
The laminar Nusselt number also depends on the heat transfer make up of the flow. For instance, it's 4.36 with a state of constant heat flux along the length of passage and 3.66 for constant wall temperature, and that's because the shape of the temperature profile is different for these different conditions. Thus, strictly speaking, even with laminar flow, the Nusselt number is not really constant, depending on different flow and heat transfer conditions.
The particular number the Nusselt number assumes is not only a function of the flow and heat transfer conditions, but also it's an artifact of the definition of h, the film coefficient. You can call these numbers a "null point," where only conduction occurs and they come about from how the definition of film coefficient is made.
With higher Reynolds numbers, inertial effects lead to turbulence, and turbulence increases as these effects become more influential. Velocity and temperature profiles also become fully developed, but they cannot be computed from first principles. Thus, the Nusselt number becomes an explicit function of <PERSON> number, and different correlations serve to provide values for different flow situations. Some correlations work better than others, and it's somewhat of an art to pick the most useful ones. 108