I don't believe the LOLER is that prescriptive but there is certainly directives that a lift must comply with under the machinery directive. In my experience you wouldn't say, have a redundant cable but you would have installed secondary protection to prevent freefall. 

The Lift Safety Directive (2006/42/EC and amended by 2014/33/EU) is one such directive which a passenger lift must conform with in order to demonstrate that it meets the machinery directive EHSR's.

"The details of the essential health and safety requirements will be the subject of a series of standards. However, certain basic principles are defined in the Directive itself. These include:"

• The lift must be automatically rendered inoperative if it is overloaded

• The speed of the lift must be controlled and kept below limits.

• Passenger lifts must have a counterweight (which can be a second lift).

• The lift shaft must be inaccessible except for maintenance or an emergency.

• A free space must be provided at each end of the lift shaft.

• The lift must be prevented from moving unless the doors are closed.

• Lifts must be completely enclosed.

• The lift must have a free-fall prevention device

• Buffers must be installed at the bottom of the lift shaft.

• Doors must be incapable of crushing persons or objects and must be suitably resistant to fire.

• The lift must be fitted with two-way communication.

• The lift must have adequate ventilation and lighting, even in an emergency.

• There should be an emergency override control system for fire and rescue.

• The lift must be clearly marked to show the maximum load and the maximum number of passengers.

• Safety components must be accompanied by an instruction manual.