Test the thermostat and get the radiator and heater core tested for flow. Wrong coolant will damage the radiator if it is aluminum. Test fan clutch and electric fan at both speeds. Change the water pump if it is suspect. Do a compression test wet and dry and a leak-down test (to rule out gasket and cracked head problems.
And you can get water in the bore from a blown head gasket without water in the oil, but you will probably have a skin of oil in the water. (Western Australia - Home of corroded cylinder heads and blown head gaskets)
Good - Fast - Cheap: Pick any two.
3 x E32 735iL (Alpine White, Glacier Blue & Kashmir Beige); 2 X E38 735iL (Oxford Green & Alpine White); E53 X5; R50 Cooper; R55 Cooper Clubman.
E38 Parts Car
E21 318i; E23 735i; E30 318i; E32 735iL; E34 535i; E38 730iL; E53 4.4i
I've just changed my wiper blades and now my reverse lights don't work. Should I change my fuel pump relay?