Unlikely that part of the ECU on the cold start side is defective as this controls the entire engine on these. Factors which would affect the cold starting and running are as follows:

Temperature sensor defective, this drops into limp home mode and limits your RPM as it puts a default setting into the system. Instead of a variable resistance, this becomes a fixed resistance and is biased towards a warm engine, so when warm it will run reasonably well.

Idle Speed Control Valve, if yours is fitted with one, it will increase the RPM when idling, this controls the air entering the engine and stabilises this when it is below its running temperature range.

As these sensors are resistances they may be affected by dodgy, corroded, or defective wiring; this alters the resistance of the circuit and gives the ECU the wrong resistance readings. This then propmts the ECU to adjust the engine management system accordingly, or if it is outside its operating range will put it into limp home mode.

Was the ECU removed from the vehicle? if not it was tested through the large multi plug called the SD connector, this is used to connect the system to the diagnostic system. This being the case, we can conclude that the fault is not necessarily in the ECU, it can be on the circuits between the sensors and the ECU.
Many people do not check the ECU off the vehicle, this gives rise to such confusion as people often condem the ECU, then the sensors, but never check the wiring or its multi plugs which are often the most likely causes of problems on older vehicles. Often a £3 can of electrical cleaner and visually checking the wiring will identify the problem instead of considerable sums spent on diagnostics and swapping parts.