Having just replaced a head gasket on one of our vehicles here in the U.S., I could probably confirm that you're problem is either a cracked cylinder allowing the coolant into the internal piston area, or a broken/damaged cylinder head gasket.
First, I'm not familiar with the Vauxhall, but all liquid cooled engines are relatively the same. If yours is a simple 4 cylinder, as an example, then there's probably only one gasket. However, a good mechanic would want to send the cylinder head (which has to be removed to replace the gasket) out to have it planed to assure a perfect fit. This adds to the cost. Ours was a V-8 so required two gaskets and cost $1800 U.S. I would expect yours to be quite a bit less.
If the cylinder or fluid temperature continues to go up after starting, don't drive it since you could really damage the engine. Here, the "bottle" as you call it, can be low or empty and still not be enough to refill the coolant system. There should be another fill port that goes directly into the engine or radiator... have someone look at it...
White smoke is always coolant, black smoke is oil...
Best of luck!