2002 Vauxhall Corsa Cooling system failure

We should be asking YOU what's leaking. You're the one who is there and can look at it. The symptoms you described don't really sound like a leaking cylinder head gasket or radiator. Look at the hoses once the engine has cooled down. Add water to the radiator or reservoir, then listen and watch for where it runs out. My hope is you find a split rubber hose. If you do, and you can post a photo of the hose, and it's easily accessible, I can tell you the name of it and you may be able to replace it yourself.

I don't know which engine you have or how to convert pounds to dollars, but a leaking water pump is common and on a lot of engines they are driven by the timing belt. At the mileage you listed you're overdue for a new timing belt. Many engines today are "interference" engines meaning if the timing belt breaks any valves that are open will be hit and bent by the pistons as they coast to a stop. That turns an expensive maintenance repair into a real expensive engine repair.

Also, when the water pump is driven by the timing belt, it is customary to replace it when the belt is replaced to insure the quality of the repair. Any mechanic who has your best interest at heart will recommend the additional part to avoid exactly what happened to you.

For a water pump to leak it has also usually developed worn bearings. That allows the pulley to wobble, and again, it that is driven by the timing belt, a wobbling pulley will chew up the belt. Sounds to me like you found a good repair shop. If that was $350.00 you'd be getting a real good deal.