JavaScript (ES6), 86 bytes

Takes input via the currying syntax, e.g., (4)('12')

n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

'use strict';

const G=n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

const submit = () => {
  console.clear();
  console.log(G(+n.value)(d.value));
}

button.onclick = submit;
submit();

<input id="n" type="number" min="1" value="4" />
<input id="d" type="text" value="12" />
<button id="button">Submit</button>

To be run in strict mode (for tail call optimisation [TCO]). If your environment doesn't support TCO it will result in a stack overflow error for primes larger than the environments stack.

For invalid inputs it will run forever.

Note: Chrome users can go to chrome://flags/#enable-javascript-harmony and enable this flag to run the above snippet with TCO support.

JavaScript (ES6), 86 bytes

Takes input via the currying syntax, e.g., (4)('12')

n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

'use strict';

const G=n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

const submit = () => {
  console.clear();
  console.log(G(+n.value)(d.value));
}

button.onclick = submit;
submit();

<input id="n" type="number" min="1" value="4" />
<input id="d" type="text" value="12" />
<button id="button">Submit</button>

To be run in strict mode (for tail call optimisation [TCO]). If your environment doesn't support TCO it will result in a stack overflow error for primes larger than the environments stack.

For invalid inputs it will run forever.

Note:

• Chrome (>= 51) users can go to chrome://flags/#enable-javascript-harmony and enable this flag to run the above snippet with TCO support.
• Safari (>= 10) supports TCO

JavaScript (ES6), 86 bytes

Takes input via the currying syntax, e.g., (4)('12')

n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

'use strict';

const G=n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

const submit = () => {
  console.clear();
  console.log(G(+n.value)(d.value));
}

button.onclick = submit;
submit();

<input id="n" type="number" min="1" value="4" />
<input id="d" type="text" value="12" />
<button id="button">Submit</button>

To be run in strict mode (for tail call optimisation [TCO]). If your environment doesn't support TCO it will result in a stack overflow error for primes larger than the environments stack.

For invalid inputs it will run forever.

JavaScript (ES6), 86 bytes

Takes input via the currying syntax, e.g., (4)('12')

n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

'use strict';

const G=n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

const submit = () => {
  console.clear();
  console.log(G(+n.value)(d.value));
}

button.onclick = submit;
submit();

<input id="n" type="number" min="1" value="4" />
<input id="d" type="text" value="12" />
<button id="button">Submit</button>

To be run in strict mode (for tail call optimisation [TCO]). If your environment doesn't support TCO it will result in a stack overflow error for primes larger than the environments stack.

For invalid inputs it will run forever.

Note: Chrome users can go to chrome://flags/#enable-javascript-harmony and enable this flag to run the above snippet with TCO support.

JavaScript (ES6), 86 bytes

Takes input via the currying syntax, e.g., (4)('12')

n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

'use strict';

const G=n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

const submit = () => {
  console.clear();
  console.log(G(+n.value)(d.value));
}

button.onclick = submit;
submit();

<input id="n" type="number" min="1" value="5" />
<input id="d" type="text" value="12" />
<button id="button">Submit</button>

To be run in strict mode (for tail call optimisation [TCO]). If your environment doesn't support TCO it will result in a stack overflow error for primes larger than the environments stack.

For invalid inputs it will run forever.

JavaScript (ES6), 86 bytes

Takes input via the currying syntax, e.g., (4)('12')

n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

'use strict';

const G=n=>(d,F=(i,P=j=>i%--j?P(j):1==j)=>P(i)&&`${i}`.match(`^[${d}]{${n}}$`)?i:F(i+1))=>F(2)

const submit = () => {
  console.clear();
  console.log(G(+n.value)(d.value));
}

button.onclick = submit;
submit();

<input id="n" type="number" min="1" value="4" />
<input id="d" type="text" value="12" />
<button id="button">Submit</button>

To be run in strict mode (for tail call optimisation [TCO]). If your environment doesn't support TCO it will result in a stack overflow error for primes larger than the environments stack.

For invalid inputs it will run forever.