Once your radiator was replaced, was the system correctly bled of air, if it wasn't you going to continually have issues.
Common leak points on the R53 are:
Expansion tank (cracked or separated seam).
Leaking O-rings in the heat Exchanger (oil cooler on the side of the Oil Filter Housing).
Hoses and or clamps.
Faulty Temp sender switch.
Faulty cooling fan
Incorrectly bled system after coolant replacement.
leaking heater core. (Damp carpet)
If your heater isn't blowing hot air, you more than likely have air in the system and need to bleed it correctly.
Don't top the system off with water, keep a bottle of 50/50 coolant/deionised or distilled water.
Topping up with water unless in an emergency will only cause corrosion in the long run.
When topping up the expansion tank, when the car is cold, the fluid level should be sitting on Min. When the car is at working temp, the coolant should be sitting on Max.
If your fluid level is above Max when the car is cold you have to much coolant in the system and when the coolant expands, it has nowhere to go and will overflow via the filler cap due to pressure and with the resulting coolant loss under pressure.
Read this article to bleed your system and then work back from there (Thermostat? cracked or leaking thermostat housing? etc etc). MINI Cooper Coolant Change (R50/R52/R53 2001-2006) | Pelican Parts DIY Maintenance Article MINI Cooper Coolant Expansion Tank Replacement (R50/R52/R53 2001-2006) | Pelican Parts DIY Maintenance Article MINI Cooper Radiator, Thermostat and Hose Replacement (R50/R52/R53 2001-2006) | Pelican Parts DIY Maintenance Article