It is known that higher codimensional mean curvature flow appears naturally in the G2-Laplacian flow of coassociative fibrations. With this motivation, we consider entire higher codimensional mean curvature flow in Rn,m of n-dimensional spacelike manifolds and prove a long time existence theorem starting from arbitrary spacelike initial data. We will see that the key to the proof is to demonstrate local spacelike gradient estimates, and to get around difficulties with cutoff functions in Rn,m. This yields new long time existence results for the G2-Laplacian flow.