I'm inclined to think that since Alice Prosser wanted other girls to have the opportunity she didn't, and the scholarships included career training/university, that she wouldn't have wanted clothes to be the stopping point for those awarded the scholarship, and that it did include uniforms, musical instruments etc. as required. But there's nothing in the text to say either way.

You're right about Nicola; she likes the family history inherent in those passed-on possessions, and that probably outweighs family liking/disliking. And she has fun with the boater (young Churchill/adsum) and it doesn't seem that it being Ann's is off-putting.