Probably you'll have several options, but I'm going to propose "mimoso".
A cuddle is
a prolonged and affectionate hug.
and "to cuddle"
hold close in one's arms as a way of showing love or affection.
So I'm think that the key to the connotation of this word if the affectionate, loving, touch.
"Cuddly" would be "pleasant to cuddle" o "wanting to cuddle".
In Spanish, "mimoso" means:
- adj. Melindroso, muy aficionado a caricias, regalón.
This adjective comes from "mimo", which is
- m. Cariño, halago o demostración de ternura.
Which I think conveys pretty well the meaning of "cuddle", and so, "mimoso" conveys pretty well "cuddly".
A "cuddly puppy" would be a "perrito (o cachorro) mimoso" (a puppy who is affectionate and likes giving and receiving demonstrations of affection).