When you lose xlogs directory – you are scrued! DB don’t start (which is actually good thing). That directory contains transactional logs.
Fortunately there is a good tooling to solve this problem. It doesn’t mean that you don’t loose data. But you can save some data. The trick is to read this pgresetxlog doc
Key is to correctly calculate all the options! If you miss-calculate, you can even loose some data. What helped me was this post. Once you manage to start the db, you just do pg_dump and load into a new DB.