In my experience, there are two major impacts. One is from the budget office for wherever you are, telling people to prepare 5, 10, 15 percent budget cut proposals. Fifteen percent can look pretty ugly!
Even before that, there could be a “hiring freeze” announcement. You may have the position authorized, you may have advertised and even be doing interviews. But, if that person has not had an offer letter, or, even better, started working ... there is a good chance that you will not be hiring anyone and instead getting by with the staff you have now.