UK Car Repair Questions? Ask a Mechanic for Answers ASAP
i would suspect the rpm signal sensor (crankshaft position sensor)
they are prone to failure especially when hot fortunatly they are fairly cheap at £25 approx so it would be worth fitting .
the coolant light will only appear if the level is low or the sensor is faulty.
As the engine check light is on it will record the fault code in the memory of the engine control module so a plug in would confirm what the problem is.