The issue here is that the Reynolds number is not a measure of "how turbulent" the flow is or will become. It is a ratio representing the relative importance of inertial forces to viscous forces in the flow.
What they teach you in your first fluids class is often that above a certain Reynolds number, a pipe becomes turbulent, but this is only part of the picture. A boundary layer transitions because there are always some level of fluctuations in the free stream. These fluctuations, if they are of the right nature, can be entrained in the boundary layer by interacting with a surface; this is especially important early in the development of the boundary layer.
Now, the boundary layer is actually a lot like a mass-spring-damper system, only a lot more complicated. In essence, it is a nonlinear oscillator which is subject to instability and resonance and all sorts of other good stuff that undergraduates often aren't taught about it. This means that when you have these small fluctuations that get entrained in the boundary layer, they are subject to the stability properties of boundary layer. A boundary layer is always unstable to some degree, so these fluctuations will tend to grow. If they grow enough, they become nonlinear, at which point the math becomes harder, but the overall concept of growth and decay stays the same. If they grow still further, they transition from a fluctuating laminar flow to the turbulent flow that everyone likes to talk about. The rate at which these fluctuations grow in the boundary layer tends to scale with Reynolds number.
Now, back to the turbulence intensity that you are talking about. Typically, the turbulence intensity that you are describing is a measure of these free-stream fluctuations that started off the whole process mentioned above. In flight, they are typically close to zero. In a wind tunnel, they are generally around 1% or so unless you have a special, low-disturbance (or quiet) wind tunnel. It is an inlet conditions that you set based on the realities of the physical situation you are modeling, not some intrinsic property of the flow.
For an open system (most external flows), this is entirely determined by the ambient conditions of what you are testing. A pipe is a special case since it is a closed system, i.e. it has no infinite boundaries. Often, when simulating a pipe, you assume it is already fully developed and therefore likely turbulent rather than trying to calculate all of that from the inlet of the pipe to where you are now, so you just prescribe a fully-developed turbulent flow at the inlet of your grid. To get the turbulent intensity there, you can scale it with Reynolds number. In reality, turbulent intensity tends to remain rather constant with Reynolds number, especially at reasonably high values of Re. Toward the lower end it is transitional, and a transitional boundary layer generally has larger fluctuations than a fully turbulent boundary layer.