1978 Other Mini Models Unusual cooling problem?

I drive a classic Mini, and have what may or may not be considered an unusual cooling problem. My engine overheats when it's idling, and my heater only blows cold air. However, as soon as I start driving, the engine temperature drops, and after about 30 minutes of driving time, the heater begins to blow noticeably warmer air (windows de-fog after about 4 minutes). Is there a common problem which causes coolant to circulate only while the engine is under load, or is it going to take more observation to figure out what's going on?