Have now found out what the "
nime box" you are talking about actually does. It's a pulse-width modulated motor speed controller set to run the radiator fans at the low speed.
It replaces the resistors that are prone to fail inside the fans themselves, and is a more elegant and efficient solution, with the big advantage that it doesn't get hot. However it is a source of 20kHz square waves, modulating 6 Amps of current at 12 -14 Volts, i.e. about 80 watts. That's a lot of pulse power going down wires meant for DC, and it is possible (though pretty unlikely) that the electromagnetic interference from these pulses (which the car was not designed for) is interfering with the function of the ABS system.
The low-tech, cruder alternative is simply to replace the internal resistors by high-power resistors attached to an external heatsink (1 Ohm 100 Watts is the spec I've seen, which is quite a beefy resistor). The main advantage of this approach is there's less to go wrong, and no EMI being generated.
Difficult to say what to do to reduce the effect. The pulsed power from the "Nime box" is applied to the fan power wiring, giving it some length of wiring to broadcast from. You could try powering-off the Nime box to see if the symptoms go, but that isn't foolproof as there may have been some cumulative effect on the ABS box. It may be as simple as moving some wiring away from the ABS controller.