Pull your spark plugs and see what they look like.
When was the last time you changed your fuel filter?

I suggest pulling the DISA valve and seeing if there is a bunch of oily residue in the intake.

If you are getting horrible mileage and no engine faults, that means something is increasing the air/fuel ratio and the computer is adding fuel to compensate.

There is either too much air, not enough fuel, something foreign (CCV blowby), or all of that going on in the chambers.

I wouldn't expect your oxygen sensors to have gone bad, but you could replace those and see what happens if you don't find cheaper, more obvious problem.

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2001 E53 3.0 5pd