I'm not sure but I have read that in some cases, therapy isn't seen as necassary because a lot of children...like my DD...will grow out of it...other might have something called cognitive behaviour therapy which for children, is presented as a fun learning experience.
Here's a link about CBT for children who are selectively mute..it's American but I think it's pretty similar in the UK
www.selectivemutism.org/faq/faqs/how-is-sm-treated
The first thing they list is what you've basically been doing already with positive reinforcement for any talking...(nothing like a parent's genius eh!) other parts of it seem to involve learning to make mouth sounds such as whispering and being rewarded for it by the therapist....
It will all be done in a fun way...I was present for my friend's son's evaluation for ASD and he had no idea he was being "tested" at all....
if your DD needed treatment, she wouldn't know it was treatment...they're very careful.
With my DD I did as you do....I never got cross or made her feel bad for not speaking...I just gently encouraged. When she was in tricky situations...such as we'd be out at the shops and someone might be friendly and ask her a question...she'd just stare at them....and I'd say "I don't think she's in a talking mood today" and laugh it off...change the subject. I tried to avoid answering for her though as I didn't want her to get reliant on me.
Sometimes, I could, if I felt it was the right situation, get her to just nod or shake her head by asking the person's question to her directly...so they'd say "Oh that's a lovely bear! What's his name?"
And I'd give her the chance to answer..which she didn't....then I'd say "It's not a secret is it DD? Shall we tell?" and she'd nod...and then I could answer for her.
It's hard but it does go OP...especially with parents who are informed and onto it. DD is 11 now and recently seems to have absolutely blossomed. She had friends all through primary...all quiet and nice kids...but now she's older, she's got more of a mix of mates and she speaks just like they do.x