Thread-safety

The implementation is not thread-safe. checkstring calls functions that mutate the internal state of the instance (naturalDate and unixTime), so if this function gets called from concurrent threads, you may get incorrect results.

You can easily make it thread-safe by eliminating the naturalDate and unixTime fields. You don't need them. These can be local variables in checkstring, and passed as parameters to getResults.

In fact, the best would be to replace getResults with dedicated functions that return {unix: ?, natural: ?} with the appropriate value filled. (See further below.)

Design

It's not great that some functions that manipulate internal state are exposed. It would be better to make them private, inaccessible from outside the class.

Alternatively, change them in a way that they will not modify the internal state.

Avoid unnecessary evaluation

Here, test2 doesn't need to be evaluated when test is true:

let format = /\w{3,9}?\s\d{1,2}?,\s\d{4}?/;
let unixFormat = /^\d[0-9]{0,20}$/;
let test = format.test(string);
let test2 = unixFormat.test(string);

if (test === true){
  this.natToUnix(string);

} else if (test2 === true) {
  this.unixToNat(string);

I would rewrite like this:

if (this.isNaturalFormat(input)) {
  return this.getResultWithUnixFormat(this.naturalToUnix(input));
}
if (this.isUnixFormat(input)) {
  return this.getResultWithNaturalFormat(this.unixToNatural(input));
}
return this.getResult();

All the functions I invented here, I suggest to write them ;-) That is, getResultWithUnixFormat takes as parameter a date in unix format, and returns {unix: ?}, and so on.

Use boolean values directly

Since .test(...) returns a boolean, you could use if (test) instead of this:

let test = format.test(string);
// ...

if (test === true){
  // ...

Naming

test and test2 don't tell much about their purpose:

let format = /\w{3,9}?\s\d{1,2}?,\s\d{4}?/;
let unixFormat = /^\d[0-9]{0,20}$/;
let test = format.test(string);
let test2 = unixFormat.test(string);

I suggest to rename them to isNaturalFormat and isUnixFormat.

Also, I would spell out nat as natural in unixToNat and natToUnix.