The author considers the problem
\[ \begin{aligned}\partial_t u-\nabla (A(x,t)\nabla u)=f(x,t,u)\text{ in }Q=\Omega \times (0; T),\;u=0\text{ on }\Gamma_D \times (0; T),\tag{1}\\ -(A(x,t)\nabla u)v =g(t,x,u)\text{ on }\Gamma_N \times (0; T),\;u(x,0)=u_0(x)\end{aligned} \] in \(\Omega\). Here \(\Omega \subset \mathbb R^N\;(N\geq 1)\) is a bounded domain with Lipschitz continuous boundary \(\partial \Omega = \overline{\Gamma}_D \cup \overline{\Gamma}_N\), \(\Gamma_D \cap \Gamma_N=\emptyset\), \(v\) is the outward normal to \(\Gamma_N\), \(A(x,t)\) is symmetric, Lipschitz continuous in \(t\) and uniformly positive definite matrix in \(Q\), \(f:\;\Omega \times (0; T)\times \mathbb R \to \mathbb R\) and \(g:\;\Gamma_N\times (0;T) \times \mathbb R \to \mathbb R\) are Lipschitz continuous. For problem (1) the full discretization scheme with Rothe-wavelet approximation is constructed. The convergence of the approximate solution to the precise solution of the initial problem (1) is proved and the full error analysis is performed. In the end of the paper the author illustrates this numerical technique by a numerical experiment for the 1-D Fisher diffusion problem.