I started an adult nursing course when I was 18, somehow I got in with no experience whatsoever. But that was back in 2005, so they may have changed things a bit since then. Our course dropped down from about 45 students to maybe 25 by the time we graduated.
Would you consider studying adult nursing and then doing a conversion course to children's. It would be an extra 18 months, but it might be something to consider? Alternatively, some adult nurses do work in children's settings. In neonatal units, the nurses are either midwives, adult nurses or children's nurses. A lot of school nurses are adult nurses, and health visitors are either adult nurses or midwives (not sure about children's nurses). And then, in some departments you will look after adult and child patients, such as in A&E or theatre.
As for what HCAs do, it does vary depending on the Trust you work in and the area you work. Where I trained, the HCAs literally provided personal care - washing and dressing, toileting, changing beds, feeding patients. Where I worked afterwards, they also did observations, some did dressings/took blood/cannulated/catheterised.
And the uni where I studies replied back too late for it to go through UCAS, but I got a place on their course