The 18-year-old, who has scored 16 times this season for the Nomads, has signed a deal until the summer of 2015.
"We've brought Rhys in after having looked at the division to see what the best young talent is there - and we think Rhys is one with huge potential," manager Malky Mackay told the club's official website.
"He'll be training with our development squad and under-18s to start with. We'll try and bleed him in gently towards the senior set-up and see how he develops, but he's certainly one with great potential for the future."
Healey added: "I'm really happy to make the move to Cardiff City.
"I'm a striker who likes to play through the middle on the heels of the last defender and I love to score goals."