Angelpoppet, I have just spent 20 minutes typing and somehow managed to delete the lot.
OK, here we go again...
Ds1 was just like this (even worse I think) at 3 and a half. He had always been a painfully shy child - toddler groups were an ordeal as he wouldn't move off my knee and would sob if another child so much as looked at him. I used to take him to the local soft play area at 9am so he could have half an hour playtime before the first other child arrived - because once they did, he wouldn't budge from my knee. We'd also had a hellish two weeks when he hadn't spoken at all (I wouldn't have believed it if I hadn't witnessed it for myself - every morning I would get up thinking "please let him talk to me today" and he wouldn't say a word, how he managed to keep it up I will never know). I'm afraid we finally resorted to threatening him to get him to talk again that time (having tried ignoring, cajoling, bribing...). we put his beloved Star Wars toys in a binbag and said we would take them to a charity shop if he didn't speak. It worked, and while I'm not proud of it, those two weeks were desperately hard. He's always had a very vivid imagination, and becomes characters - we had a Tom Kitten stage (no speaking, just meaowing), an Iron Giant stage (no speaking, just lots of stiff walking around in a robot fashion), an R2D2 stage (much beeping and squeaking, no speaking).
Anyway, that's the background - don't know if it rings any bells with you. When he was 3.5, he had been at nursery 2.5 days a week for 7 weeks when his teacher called me in to express her "serious concerns" about him. He didn't speak at all, not to staff and not to other children. He did make funny little noises, but nothing approximating words. She was particularly concerned that he didn't interact with the other children at all - he just wandered round aimlessly and didn't play. He also wouldn't make eye contact. To say I was devastated would be an understatement - he had been coming home, telling me about his day, singing me songs they had learned. It slowly dawned on me that the teacher was gently suggesting that my son was autistic - I am quite sure she had me down as a Mum in serious denial as I insisted that he did talk at home, that he had a vast vocabulary, that he did play with other children and make eye contact.
I walked home in a daze and just cried - I am quite shy, and I had suffered from PND when ds1 was born, so of course I convinced myself that I had somehow damaged him.
Over the next half term, up til Christmas, I worked on him at home ("you must SPEAK so the teachers know how clever/kind/funny you are" etc.) and the staff tried to as well, though with limited success. He finally spoke on the last week of term - "can I be the handsome prince now please?" in a song they were singing, his teacher was nearly in tears when she told me and I was going "I told you! I told you!"
Then, just as I felt we were making progress, we moved house and area and he had to start at a new nursery. I went in with his record, which included the information that the Nursery had been considering starting the statementing process (statement of special needs) and talked to the staff and let them know how worried I was that he would withdraw completely into his shell again just as he was starting to improve.
When I picked him up after the first day the teacher told me he had announced "My name's S and I live in a big new house [we don't!] and my Daddy works at the hospital" !
Since then he has slowly and steadily improved. he was still very shy and withdrawn in Reception (they were more concerned than I was - they hadn't seen him when he was REALLY shy and withdrawn!) then he suddenly came into his own in Year 1, in fact his teacher looked surprised when I asked if she would class him as a shy child. He's almost 7 now, in Year 2, and thriving - he is popular and sociable, doing well academically, and his teacher thinks he is fantastic.
So my advice would be to keep on at them in playgroup (I think it can be easy for them to overlook a quiet child, after all they are no trouble), keep talking to your dd, not in a pressuring way, but in a gentle, encouraging way and hopefully your daughter will gain in confidence, as my son has done. My son has done much better in the smaller, more structured environment of a classroom than he ever did in a playgroup/nursery free-for-all environment. I also think that the fact your dd does play with and talk to other children is a huge positive in her favour - she really does just sound very shy around adults, rather than any serious underlying problem.
Don't let her know you are worried, and don't try and deny her shyness - I have always told my son that the nicest people are a bit shy (I believe that too!) but not TOO shy. I really hope she finds a bit more confidence soon -I know how hard it is as a parent when you know your child is capable of more than they are showing.