Folks go about the training in different ways, but in general, it's becoming more of the norm to start receiving the neuro training concurrently with the general clinical psych training. Whatever you do, just don't skimp on that latter portion (i.e., the general clinical psych training); neuropsychologists are still psychologists first, and our training in psychology is often what sets us apart from our colleagues in various settings and allows us to make unique contributions to patient care and program evaluation/outcome.
However, if you get no neuro training in grad school, then the cards will be stacked against you a bit, and you'll definitely want to get a peds neuro-heavy internship in order to be competitive for postdoc. If you get little-to-no neuro training in both grad school and internship, then your odds of obtaining a quality peds fellowship are going to be pretty small. Having a child focus from the start shouldn't automatically make getting a neuro internship difficult, it'll just make the more adult-oriented programs wonder if you're going to be a good fit for them. Personally, if I knew I were going the peds neuro route, I'd look to start getting both peds and neuro training in grad school, although would also want at least some adult experience.