The oil would not have a direct effect on the cooling system.
If your cooling system is puking water then you have a significant problem.
Best case is that the radiator cap has failed and is not holding pressure. Make sure you have it on tight, and that it isn't damaged. Perhaps you can borrow one from someone else to see if it is the cap or something else.
If the cap holds pressure then it means that something is causing the pressure in the cooling system to go up above the release point of the cap. The two likely scenarios for this are:
1) Head gasket leak that is allowing cylinder pressure to leak into the cooling system.
2) Something is causing the bike to overheat.
Lots of reasons why it might be simply overheating. Radiator could be damaged, pinching off enough of the tubes such that you just weren't getting enough cooling.
Dirt or other debris blocking the radiator
Water pump has failed
Engine failure of some sort that makes it run really hot (too lean, timing off, valves badly adjusted, etc.)
You are simply riding it too slow and lugging the engine.
Rod