Personally I think that the restricted product description should be avoided.  It is best to define $\widehat{\mathbb{Z}}$ to be the inverse limit of the system of all quotients $\mathbb{Z}/n$ (without gratuitously factoring $n$ as a product of primes) and then put $\mathbb{A}=(\mathbb{Q}\otimes\widehat{\mathbb{Z}})\times\mathbb{R}$.  Now the adeles for any number field $K$ can be defined as $\mathbb{A}\otimes K$.  The connection with primes/valuations for $K$ should be a theorem, not a definition.