If I can't get my emergency passport today, I'll cancel the conference trip. I just can't justify BA's change fee.
So the best case scenario is: I get my emergency passport this afternoon, fly back to Boston and immediately hop on my original flight to London and Vienna. I'll attend the conference then wait in Vienna until I get my full passport and then apply for the US visa.
If I can't get my emergency passport today, I don't know if I'll overnight in DC or head back on the flight to Boston I just booked.