The Brexit has not been put into place yet, you know that right? Only the vote to exit has happened.
The presumptive cast of Lazarus is not the only American actors working in the UK or on the West End. There are already procedures put into place for that, and nothing has changed yet. (Except the value of the pound.) I don't work with actors contracts, so I don't know the answer. But again, nothing has changed yet with the points you raise.
"Plus the whole thing is a backlash against foriegners. You are bringing in a non-Brit to play David Bowie, a British icon."
He's not playing David Bowie...American & UK Equity do exchanges all the time.