Tried that - but they're not having it... and they have my car...which is now broken and is needed for work. Should I let them fix it, pay under protest (on my credit card), then claim back ?

This is surprisingly common with modern alloy engines. So is just one of those things.

The only thing the garage did wrong is not to warn you this could happen when you booked the car in.

I am afraid that you are going to have to suck it up and pay the bill. You might get a goodwill discount but that is about it. If it is anyones fault then it is down to BMW surely.

No it's usually the threaded outer part of the glow plug that will come out, but as it unscrews it shears the very thin tip off which remains stuck in the cylinder head.

The best chance of getting glow plugs out of an aluminium head without breaking them is to start to unscrew them with the engine hot and idling, if they are easily accessible, the heat and combustion pressure are then helping to loosen them without breaking the tip off.

There are various tools and tool kits available to extract broken glow plugs without removing the head, but it's a bit of a hit and miss job and can damage the cylinder head.