Jtm may be right about the select mutism but there are a few other possibilities. Maybe she is just shy around adults and doesn't want to speak until she is sure it is safe for her to speak. Children will speak with other children because it is safe. They are in the same mind frame, usually around the same size, etc. We teach our children not to speak to strangers and then we stick them into a school setting which is warranted, but then our children probably go to school thinking to themselves" If I'm not suppose to talk to strangers, then why are mom and dad putting me into this strange place full of strangers?" " I'm just not going to speak to them so that I don't get into trouble at home!" Children really are smart. Maybe you should take her to see a doctor to make sure her hearing is fine and her overall health. If children cannot hear right, then this will delay their speech. Does your 3 yr old speak at home fluently? If she does, then you and her caregivers needs to constantly be speaking with her about school and how she is suppose to participate by answering questions and speaking with the teachers. Maybe she just needs reassurance that it is okay for her to speak to her teachers. Maybe using a reward system for having great days for participating at school would be a great idea. Let her make a sticker book. For every time that she does something good, then she shall receive a sticker. This should work well for your 6 yr old as well. Maybe a little competition between the two girls would help both of them to do things for you or each other. By saying competition, I mean if you were to challenge the girls to help each other get better for school. For every time the older one helps the younger one to maybe read a book or practice something then she gets a sticker, and for every time the little one does what she is suppose to in school or at home then she gets a sticker, including speaking with the teachers. Do this for a while and you'll see that it will work. You have to find something that the children like and will work for. Stickers are pretty inexpensive. This system can work for house chores as well. I would definitely take her to the doctor and get everything checked out for sure though. Make sure there is nothing wrong health wise with her. Good luck.
I suspect your child suffers from Selective Mutism (although from your description, it appears to be a relatively mild case). The best site on the internet (in my opinion) is "selectivemutism.org" where you can find information and treatment options (probably only intervention may be required for your child). I might suggest you start by reading the FAQ's.
I assume you live in Great Britain. If so, you are lucky. Two of the world renown authorities of this disorder live in England - Maggie Johnson and Alison Wintgens. There is a website called "SMIRA - Selective Mutism Information and Research Association" based out of the UK which should offer lots and lots of help. It might be advisable to purchase their book (expensive) or borrow from the public library. SM is an anxiety disorder whereby the prognosis is excellent for early treatment. I wish you the best ....