Westies have a so-called double coat, which means a short, soft undercoat and a firm, stiff topcoat. The coat will not moult naturally and the older coat has to be pulled/plucked on a regular basis. For obtaining a good dense coat this should be done at least every 3 months.
A puppy's coat is plucked when the puppy is about 10 to 14 weeks old. Only then will the tough topcoat, that the adults have come through.
The coat needs to be brushed regularly to keep it in good condition. You should let your puppy get used to its (preferably daily) brushing from the moment you have him/her. This is a good opportunity to check its ears and mouth. They generally do not like this and the resistance can be fierce. Some puppies even turn to biting with their razor-sharp teeth, which can hurt quite a bit. This is of course not allowed and must be corrected in the right way, namely:
a short, powerful "no" accompanied by a pinch on the side of the neck (they really have to react). As the puppy grows older the resistance will subside. It will take up to 6 months before the puppy will seize its protests at its worst, but you should not give up!