OK, a 1999 US E39 uses the following speed pulses:

Front left-wheel: DFAVL signal, yellow/red wire
Front right wheel: DFAVR signal, yellow/white wire

Nav:
DFAVL pin 10 of X1312 (violet) connector
DFAVR pin 11 of X1312 (violet) connector

ABS control unit:
DFAVL: pin 35
DFAVR: pin 18
The ABS control unit is attached to the ABS pump in the engine bay

The common fault with these ABS units is that internal solder connections break due to the heat and vibration and ageing inside the engine compartment.
As your car is 10 years old, this is likely to be the problem.
A good reliable solution is a brand new replacement ABS control unit.
Some companies in the USA fix the old ABS units and provide guarantees, but please do not attempt to fix it yourself unless you are an electronics engineer with experience!

ABS warning light wiring:
ABS unit, pin 17, grey/black wire, goes to...
...low cluster: pin 16 of X11175 (26p black connector)
...high cluster: pin 15 of X16 (18p black connector)
High cluster has the check-control display that shows eg light failure messages in plain text
Low cluster has the pictogram with a car picture showing doors open etc.

Print out the above and go back to your BMW mechanic. He should readily be able to diagnose the problem.