HI and welcome, Sixpence :)
I have to say that your Westie is quite unusual (to me) in her pickiness. Most times westies are appetite surrounded by dog and eat way too much....
Personally, I think she's been pampered and indulged in this matter by her old owner. It is normal for dogs/puppies to go off their feed during the first weeks in a new household. Unfortunately, many owners rush to try to find the right food and give the dog all sorts of tempting delicacies which the dog starts to take for granted, then become even more fussy. :rolleyes: A vicious circle, eh? ;)
My suggestion would be to put whatever food you want her to eat down. Direct her to the bowl and give her 5 minutes to start eating. Don't let her grab a bite and then wander off with it. As soon as she leaves the bowl, or she decides she'd not have her food...put it up.
Whenever your next set time is to feed her...do the same thing all over again. It could take a week + to do this. But a healthy dog will not starve herself. If you give in, that's it. She'll know and Westies LOVE to be in charge ;)
You said you had another dog. There could also be some interaction in the room where you feed with posturing or stress by one another for whatever reason. Food is one of those 'trouble areas.' You might not be picking up on it due to extreme subtility. Separate them or at least have their backs to one another. When one dog finishes eating, have he/she leave the food area completely. It will become a habit after a while.
good luck and enjoy your new 'puppy' :D :D
regards,
toodles