When reparing the idle speed fault in my E300D I found that the ELR (idle control unit) is the first junction from the rpm sensor, it has a pin called TD - which then goes out to other bits like the klima and the gauge.

The A/C relies on rpm information to reference it to the rpm sensor on the compressor (detecting belt slippage if any).

If the ELR is inoperative (no idle speed control) then the rpm wont work and neither would anything related to rpm sense. The usual culprit is the OVP as it switches on everything, but sometimes its also the fuse no. 6 in the fuse box (old flaky fuses cause such)

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2012 BMW X5 (Beef + Granite suspension model)

1995 E300D - The original humming machine (consumed by Flood 2017)
2000 E320 - The evolution (consumed by flood 2017)