in the case of a e90 diesel, as i assume yours is..
the first cat or pre-cat is incorporated in the dpf , its purpose is to warm up fast and control startup emissions.
the second or primary cat is the bigger one that has more capacity, but heats up slower.
the lambda sensors after the cats will tell the ecu that they are missing, or that something else is wrong. you should have cat foolers in place to fool the ecu, the ecu will not adapt to a missing or non functional cat.
white smoke aka burning oil, can be caused by oil entering through the intake, via the breather (pcv), or compressor side of the turbo, or engine itself via valvestem seals or piston rings.
futher it can be from the exhaust side of the turbo, where a worn turbine seal might be letting oil through due to less backpressure (due to removal of the cat's restriction)