Variable-order time-fractional diffusion models describe systems whose memory intensity evolves with time, but their numerical treatment is complicated by changing history kernels, global nonlocal boundary coupling, and the growing cost of memory terms. This paper develops a fully discrete implicit finite-difference scheme for a variable-order Caputo diffusion equation subject to an integral nonlocal boundary condition. The time derivative is approximated by a variable-order L1 formula on a uniform time grid, while the spatial operator is discretised by second-order central differences. Eliminating the nonlocal boundary value produces a rank-one modification of a tridiagonal system, which is solved efficiently by the Sherman-Morrison formula. Under stated regularity and discrete well-posedness assumptions, the method is unconditionally stable and satisfies an error estimate of order O(h^2+Δt^(2-α_max ) ). The manufactured-solution tests reproduce the expected temporal rate, satisfy the nonlocal boundary relation to machine precision, and quantify the loss of accuracy caused by short-memory truncation. For long simulations, fast history compression is preferable to aggressive truncation.............. Keywords: ................variable-order Caputo derivative; time-fractional diffusion; nonlocal boundary conditions; L1 scheme; stability; convergence; fast memory.