Doesn't seem like you did anything wrong on BA.com, usually tkts are issued instantly. I wonder if going in immediately to add passport details (instead of adding while in the booking/payment flow) caused the ticketing process to get a 'simultaneous change' and not be able to complete? Shouldn't have, but might explain it. Whatever it was that failed, it just wasn't fixed by the time you got to the airport due to the lateness of the booking (still not your fault).
Would have been nice if the LIN staff had been more helpful, calling the Bremen call centre internally themselves and getting through, and things resolved, much faster. With no ticket counter (and quite possibly no actual BA employee on site) they told you to call instead. Sadly, that's a failing of the LIN contract staff and/or BA for not giving airports sufficient resources for when things do go wrong.