With OTL automotive technology, they would drive to Jutland, then perhaps take a ferry across. Or alternately, drive to an airport, fly, and then rent a car.
In ATL automotive technology, they would drive to a local station. There they would disengage the engine, attach their coach to a train pulled by an engine as if it were on rails, and ride a kind of segmented bus composed of their own coach and other coaches to Jutland. There the train of coaches would board a ferry. The engine remains behind after it loads the train on the ferry to make a return trip. The train of coaches travels on the ferry to Oslo, or for the sake of argument, Christiania. There it is unloaded by an engine specific to the purpose, and a service there rents engines, attaching different ones to each coach. The family then drives north on its own scenic tour.
What's gained is some of the efficiency of train travel, with one engine pulling multiple loads in a kind of combination train/tractor trailor/bus. But you also get the versatility of the automobile, with the family on vacation reaching its destination able to split off and travel in their own direction, not having to follow a set of rails, meet a schedule, or sleep while on a moving train (which Mutti just can't do).
The real advantage of a train is the steel on steel wheels. Here, the ONLY advantage you have is a possibly more efficient puller - but then you have to be able to hook the coaches up in series, which seems like unnecessary extra complication.
I don't see this as being at all practical, sorry.