We upgraded my friends R53 with R56 rear suspension. This is the 4th Gen 1 that we've done this to and have not had this problem before.
We used all genuine Mini parts from the dealer except 1 hub came from the local euro-parts shop. I don't know the manufacturer of the one hub.

Upon test drive, got the abs, low tire and triangle lights, so I hooked up the scanner and the right rear wheel speed was zero. WE HAVE SWAPPED EVERYTHING AND STILL READS ZERO. Ohmed the wires and they are good.
We jumpered the right sensor to the left plug and we got a good reading. We even slaved in the original Gen 1 trailing arm and it worked fine. Lastly, we pulled the Gen 1 abs sensor and stuck it in the whole and it reads correctly when rotor was spun.

So we ohmed out the Gen 1 and Gen 2 sensors and the Gen 1 is 4M ohms and the Gen 2 is 0.4M ohms. We just can't figure out why the right doesn't work when fully assembled. (the Gen 2 sensor was too short to try it in the Gen 1 trailing arm)

Any suggestions? Again, these are brand new OEM parts. We looked at the hub mating surface and measured the sensor locations with a ruler and they seem very close.

 

I forgot to mention we did a load light test on the wiring and it was good.

 

Tried a 3rd ABS Module and everything works now.

 

Before reading everything... I was gonna ask 'what are the odds something has gone bad in the swap???'

 

I had to use R53 sensors on my car with R56 rear trailing arms. I bought some adapters on etsy (just search for them) that allow the R53 sensors to match the R56 sensor diameter. Also, make sure you're using OEM Mini sensors, not the OE supplier ones. I chased this for weeks before buying Mini sensors and now no more issues