To the best of my knowledge there are no longer any passport checks on exiting the UK (only occasional spot checks for ID purposes and security pretence). Therefore leaving is possible except that there will be airline check-in hurdles and without a passport you are likely to be refused check-in - airline requirement including destination country's restrictions. You must be able to prove who you are, that you have a valid passport (and any required visa) to enter the destination country - without that no airline is likely to accept you. You best bet is to go to your embassy in the UK and obtain a replacement document - this depends on their rules and regulations. Without a travel document you will likely not be able to travel.