Many of the children who audition for London musicals have agents who apply on their behalf, but an agent isn't necessary and, for the big West End shows and tours in particular, applications can come directly from parents.
The two main casting directors involved in children's casting for West End shows are Jo Hawes and Jessica Ronane. Both have websites where you can sign up to receive news of forthcoming auditions.
The "must live within the M25" stipulation is pretty standard for London shows, but in practice children in these shows often come from quite far outside London - generally as long as travelling time from home to the theatre is no more than about an hour, it's fine.
A lot of shows have height restrictions, so it helps if a child is small for their age.
Most auditions involve a initial round where large numbers of children are seen, often in batches, and then a smaller number are recalled to the next stage. There may be just one recall stage and then a final decision is made, or there may be several recall rounds culminating in "finals" involving just a handful of children. Sometimes the children to be recalled are announced at the first round audition, sometimes an invitation to recall comes a few days later by email.
For some auditions, children are given a song or some script to prepare in advance and perform in the first round. For others, children are asked to prepare a song and/or monologue of their own choice - if this is the case, usually best to pick something the child knows well and is comfortable with, and which suits their age, personality and voice.
If your DD really wants to give it a go, I personally think there is very little harm in trying. The audition experience for West End shows can be great fun and very positive for children. Casting directors (particularly the two I mentioned above) are usually very mindful of the fact that they are dealing with young children, and try to put children at ease as much as possible because they to see the best performance the child can offer.
In my experience, rejection is handled sensitively. I've seen casting directors remind children not to loudly celebrate being asked back to the next round because they need to consider the feelings of children who haven't got through. It is absolutely nothing like the auditions you see on TV for things like X Factor! My DS has done a fair few auditions and usually comes away having had a great time and made a few new friends.
It is important how a child is prepared mentally for auditions and the possibility of rejection though. Numerically, the chances of being cast are small, so treat it as a fun and interesting opportunity to learn something about how the theatre works. Children are often unsuccessful at auditions for lots of reasons that are completely beyond their control (and have nothing to do with how talented they are) - such as their height. Mostly, not being cast boils down to not being quite right for the role (and this is the line I take with my own DS).