Grundig GWD59400C goes into safety mode while drying.

Hello there. Presumably it is plumbed into the cold water supply and the cold water supply is still turned on when using the drying cycle? I'm also assuming that the washing machine works perfectly okay on wash cycles? If so, it sounds as if there is a problem only with the drying section. If it says safety and then shuts down, it may well be overheating.

When on a drying cycle, the heater should operate, the drum should turn, and a dryer fan should be constantly running to blow the hot air through the laundry. There should also be a trickle of cold water that constantly runs into a chamber at the back so that the steam generated hits this flow of water and condenses back into water in order to be pumped away.

Any one of those processes will cause a problem. The problem is we don't know which one it is, so I can only suggest you observe it carefully to ensure that all of these things are happening. And as it has been transported I would disconnect it from the mains, take off the lid, and ensure that nothing has come loose including of course wires to the fan.