Does accuracy of the numerical solution of a PDE depend on the scaling of the variables?

Question

I am trying to solve the 1D heat equation numerically. I want to solve it with a time-step of 2 seconds. Should time step always be less than one? i.e. should $\Delta t \leq 1$? Suppose our original equation is : $\begin{equation} \frac{\partial u}{\partial t}=\alpha^2 \frac{\partial ^2 u}{\partial x^2} \end{equation}$ If i make the equation dimensionless by introducing $\hat{t}=\frac{T}{T}$,$\hat{x}=\frac{x}{L}$, and $\hat{u}=\frac{u}{U_*}$, where $T$,$L$,$U_*$ are the total simultation time, length of the rod and ambient temperature respectively, we obtain the dimensionless equation as below: $\begin{equation} \frac{\partial \hat{u}}{\partial \hat{t}}=\frac{\alpha ^2 T}{L^2}\frac{\partial^2 \hat{u}}{\partial \hat{x}^2} \end{equation}$. I am unable to understand if there will be a difference in the numerical errors (due to truncation of the Taylor series) when we apply a numerical method (say Backward Euler) for the first equation and for the second equation? Will the solutions differ if we scale back our solutions of the dimensionless equation back to the original variables?