Replace the switch with an array:

static const char * numbers_text[] = 
{
  "Zero", "One", "Two", "Three", "Four",
  "Five", "Six", "Seven", "Eight", "Nine",
};
//...
    cout << "Enter a whole number between 0 and 9" << endl;
    cin >> numinput;
    if ((num_input >= 0) && (num_input <= 9))
    {
        cout << numbers_text[numinput] << "\n";
    }
    else
    {
        cout << "Unsupported number.\n";
    }

The array or lookup method avoids any issues with a switch statement.

Fewer lines of code, fewer chances of injecting mistakes.

You can't return int if int main() return string or simply said by the compiler:

no suitable conversion function from "std::string" to "int" exists

this is how you fix it:

#include <iostream>
#include <string>

using namespace std;

int main() {
    int numinput;
    string numberWritten;

    cout << "Enter a whole number between 0 and 9" << endl;
    cin >> numinput;
switch (numinput) {
    case 0:
        numberWritten = "Zero";
        break;
    case 1:
        numberWritten = "One";
        break;
    case 2:
        numberWritten = "Two";
        break;
    case 3:
        numberWritten = "Three";
        break;
    case 4:
        numberWritten = "Four";
        break;
    case 5:
        numberWritten = "Five";
        break;
    case 6:
        numberWritten = "Six";
        break;
    case 7:
        numberWritten = "Seven";
        break;
    case 8:
        numberWritten = "Eight";
        break;
    case 9:
        numberWritten = "Nine";
        break;
    default:
        numberWritten = "Not a number I know";
        break;
}
    if(numinput >= 0 && numinput <= 9) {
        cout << "The number you entered is " << numberWritten << endl;
    } else {
        cout << numberWritten << endl;
    }
    cout << "The number you entered is " << numberWritten << endl;
    return 0;
}

I apologize I forgot to test the default now it works perfectly.

The switch will never arrive on the

return number

As it's still inside of the curly braces. In the switch it'll stop on the default.

Your function is int main and you're trying to return a string with numberWritten.

This way it works

#include <iostream>
using namespace std;

int main()
{
int numinput;
string numberWritten;

cout << "Enter a whole number between 0 and 9" << endl;
cin >> numinput;

switch (numinput)
{
case 0:
    numberWritten = "Zero";
    break;
case 1:
    numberWritten = "One";
    break;
case 2:
    numberWritten = "Two";
    break;
case 3:
    numberWritten = "Three";
    break;
case 4:
    numberWritten = "Four";
    break;
case 5:
    numberWritten = "Five";
    break;
case 6:
    numberWritten = "Six";
    break;
case 7:
    numberWritten = "Seven";
    break;
case 8:
    numberWritten = "Eight";
    break;
case 9:
    numberWritten = "Nine";
    break;
default: "Not a number I know";
}

cout << numberWritten << endl;

return 0;
}