@sittingpretty Welcome! You are indeed in the right place.
In essence, both NYMT and BYMT audition across the country to find talented young people to take part in their summer shows. If you get cast in a show you will have at least one (possibly two) residentials for rehearsals, and then several performances of the show at the end. The standard of the productions is excellent for both organisations, and I say this not to intimidate prospective applicants, but to emphasise what a great experience it is for the young people involved.
In addition, both organisations run some courses and workshops for young people who may not wish to do a show, or perhaps need to work on some skills first. How and when these are organised varies between the two organisations. I believe that with BYMT you can just sign up, whereas NYMT holds auditions and uses that process to place the young person on the best kind of course for them. Thirdly, NYMT places some applicants in 'new musical workshops', which give a young cast the opportunity to experience being a part of the creation of a new show, and writers of new UK musicals a chance to see their ideas come to life. The other main difference is that BYMT has a repertoire of its own shows, whereas NYMT offers a mix of established, well-known shows (eg Fame and Carousel, which have been announced for next year), new commissions (Lightning Boy, which was successfully workshopped last summer and has been given the green light for a full production next summer) and occasionally shows that don't get seen much in the UK (in recent years, a stupendous production of The Hunchback of Notre Dame in Manchester Cathedral, and for the next summer the very first UK production of Tuck Everlasting).
From here on I'm going to talk purely about NYMT, because that's the one I know about. On the website you'll find a list of audition dates and venues. You choose whichever suits you best, and you pay a fee of £45 for a full day audition workshop with professional creatives. These workshops are really fun and friendly, and in my experience the young people come out buzzing. Your young person just needs to prepare a musical theatre song to sing, and take along sheet music - the acting and dance is all done on the day. Don't worry about the dance! Lots of auditionees are not dancers - they just want to see the young people moving and performing and enjoying it. The trained dancers are more likely to end up in the more dance-heavy shows, but previous dance lessons are not required. If your DD loves to sing and act, it's definitely a day she will enjoy. From there she may get offered a course, or she may be invited to a casting weekend session to determine whether she is offered a role in a show or a place in one of the new musical workshop casts, or an advanced skills course. NYMT has a commitment to being inclusive, so yes, 1000 people or maybe more will audition, but everyone who auditions will be offered something. Of course, if what you're offered doesn't suit you, there's no obligation to take it up, but the opportunity is there to be part of the NYMT Company.
There isn't a 'junior program' as such, because all the shows tend to need a range of ages, from 10-23, but I do know that for the courses last summer they split into smaller groups by age group.
The cost will depend on what your DD ends up doing. I'm linking here to the fees page, where you can see what courses, new musical workshops and shows cost. nymt.org.uk/take-part/actors/
There is a bursary scheme for cases of need. Obviously, most of the experiences offered involve a residential, and they all involve the work of brilliant professional creatives, so yes, they do cost, but I know NYMT has actually tried to reduce costs year on year - and if you work them out as a daily rate, and consider that your young person is getting bed and board, excellent pastoral care, tuition from the very best creatives and an experience that they will probably treasure for ever, I actually think it's very reasonable.
I hope that answers some of your questions. If you have more, don't hesitate to ask! I'd definitely encourage your DD to book an audition - it's a wonderful organisation, and if nothing else she will have a really fun day and meet some like-minded young people in her area.