Consider a d-dimensional system of stochastic differential equations with m independent diffusions where the drift and diffusion coefficients are not globally Lipschitz continuous but instead only locally Lipschitz and together satisfy a Khasminskii-type montone condition. It is known that the explicit Milstein scheme fails to converge for such systems when applied over a uniform mesh. We construct an adaptive mesh that responds to the local behaviour of solutions by reducing the stepsize as solutions approach the boundary of a sphere, invoking a convergent backstop method in the (rare) event that the timestep becomes too small. With such a mesh, order-one strong L2-convergence of the scheme can be ensured, even when the diffusion coefficients of the SDE are non-commutative. We examine how this adaptive strategy can be modified to allow for the discretisation of systems of SDEs perturbed by a Poisson jump process independent of the perturbing diffusion processes, without loss of order, as long as the independent jump times can be pre-computed and included in the adaptive mesh. We will demonstrate the use of an adaptive scheme in the modelling of stochastic telomere length dynamics arising from human cellular division over time.