JAVA Solution. Easy to understand. All test cases passed.

public static List waiter(List number, int q) {

    Stack<Integer> numberStack = new Stack<>();
    for(int no : number) {
        numberStack.push(no);
    }

    List<Integer> primeNos = getPrimes(number);

    List<Integer> answers = new ArrayList<>();
    int curr = 0;
    while(q != 0) {
        Stack<Integer> a = new Stack<>();
        Stack<Integer> b = new Stack<>();
        while(!numberStack.isEmpty()) {
            int top = numberStack.pop();
            if(top % primeNos.get(curr) == 0) {
                b.push(top);
            } else {
                a.push(top);
            }
        }
        curr++;
        numberStack = a;

        while(!b.isEmpty()) {
            answers.add(b.pop());
        }

        q--;

        if(q == 0) {
            while(!a.isEmpty()) {
                answers.add(a.pop());
            }
        }
    }

    return answers;

}

public static List<Integer> getPrimes(List<Integer> number){
    List<Integer> primeNos = new ArrayList<>();
    int no = 2;
    while(primeNos.size() != number.size()) {
        boolean flag = false;
        for(int i=2; i<=Math.sqrt(no); i++) {
            if(no%i == 0) {
                flag = !flag;
                break;
            }  
        }
        if(!flag) {
            primeNos.add(no);
        }
        no++;
    }
    return primeNos;
}