For UK passport holders (ignoring anything like work, Bulgarian family etc.), the country is red listed for us, so we can't really go there (at least from the UK). Not sure if US or EU passport holders are different? Might have to check depending on how the Bulgarians are treating your country.
In the past, it has been a case of getting off the plane, and possibly hanging around for the next plane I believe.

BA have cancelled a few flights later in the year, so Fri-Mon? only which has messed up a couple of my trips.