The vanity basin waste:
If the vanity basin is opposite to an outside wall you can use a P-trap and go directly out through the wall and drop down into the gully dish or into the existing waste pipe. Using a cleaning eye bend where it comes out of the wall. Or if the existing waste is still accessible under the floor it can either be transferred into the wall cavity or brought up to a P-trap.
Or an S-trap can be used and taken downwards through the floor. Also if you have a concrete floor and your new vanity cupboard trap position is to the left or the right but still under the cabinet the waste pipe can be cut just above the floor and redirected along the floor on top of the concrete underneath the toe space. Than brought up closer to the trap.
If you are shifting the vanity basin to an internal wall position. You can either go into the wall and around inside the wall until reaching the outside of the bathroom or run the waste pipe under the floor to the gully dish or lower with the use of a four-way riser or if already a PVC riser to a gully dish placing a single 2-inch (50 mm) boss strap saddle on the outside of the riser below the dish by drilling a 2-inch (50 mm) hole and gluing on the saddle to take the new waste pipe.