You can set route path dynamically when you get timestamp changed, at that time you can use like below

<Link to="/${yourdynamicpath}">{Title}</Link>

Edit: I notice you want to add a check to update it without refreshing, so I'd adjust the functional component to store the first date in a const [currentDate, setCurrentDate] = useState(new Date()) and within the useEffect update that with an interval e.g.:

useEffect(() => {
  const interval = setInterval(() => {
     setCurrentDate(new Date())
  }, 1000); // set this as you wish on how responsive you wish it to be
  return () => clearInterval(interval);
}, []);

Depending if you already have a date formatting library installed the exact implementation could differ, but for example if you use date-fns you are provided lots of helpful utils like:

compareAsc(dateLeft, dateRight)

which can be used like so to compare dates, we compare todays date new Date() to december 1st new Date('1/12/2021'), if the first date is before the second date it returns -1, so the condition is true and we show the coming soon, if its after we return 1, the condition is false and then we show the store:

 return (
  <>
    <Menu />
    <Switch>
      <Route path="/" exact>
        <LandingPage />
      </Route>
    </Switch>
    {compareAsc(currentDate, new Date("1/12/2021")) === -1 ? (
      <Route path="/coming-soon" exact>
        <ComingSoon />
      </Route>
    ) : (
      <Route path="/store" exact>
        <Store />
      </Route>
    )}
    <Footer />
  </>
)

Not that it's necessarily recommended to pull a package for everything - but for dates these provide lots of handy utils so getting used to all that they have to offer can be useful if you haven't seen them yet!