I’m not a lawyer, and so this doesn’t constitute legal advice, but:

To the best of my knowledge, there’s no reason that a deed poll for a child under the age of 16 couldn’t be signed by the parents and that would be sufficient: no need for a solicitor, just as with an adult deed poll. A commenter at https://danq.me/projects/freedeedpoll/#comment-85003 has provided instructions on modifying my template to do that, if you’d like. Note that understanding who the “parents” are is really important for a child’s deed poll: it’s NORMALLY all of the adults named on the child’s birth certificate (regardless of who has custody or guardianship), but in cases of adoption, re-marriage, divorce etc. can be more complicated than that: for example a parent who re-marries usually needs the birth parent’s authority to change their children’s names, even if the birth parent is otherwise estranged from them!

I’ve spoken to people who’ve successfully used the information at the link in the paragraph above to change their child(ren)s names via deed poll, so it certainly enjoys mainstream acceptance. But I’m afraid you’re mostly on your own. However, I’d be very interested to hear about your experience if you go down this route, and possibly to share what we learn with others in your position: get in touch and let me know!

Good luck!