Hearing aids and loop systems
Hearing aids in general should not be subject to any interference from power-frequency fields.
With loop systems, and hearing aids when switched to the "loop" setting, there is a risk of interference. The loop systems are designed to have a very low sensitivity at 50 Hz to avoid these problems, and there's a British Standard (60118-4) that sets criteria to ensure that interference is avoided, but some sensitivity to low-frequencies is necessary - that's how they work - so there's always a possibility of some minor pick-up in extreme circumstances.
The British Standard on Hearing Aid EMC (electromagnetic compatibility), 60118-13, says that it is not possible to set immunity levels at 50 Hz, but recommends following the privisions for loop systems of BS60118-4.
Cochlear implants have an internal, implanted, component - the receiver and the electrodes - and an external component - the microphone, processor and transmitter. Typically, the external components are held on to the head by a static magnet and transmit to the implanted receiver using radiofrequencies.
In general, it appears that static magnets should be avoided - because they can interfere with the magnets used to attach the external receiver - and anything that injects currents in the head should be avoided because of the risk of damaging the implant. Static electricity was recognised a problem in the past but may be becoming less so. But sources of EMFs do not pose any risks, although they may produce a noise through the implant.