No UK government body issues a licence to operate plant or a forklift, and no statute requires one. Below are straight answers to the questions that actually come up, followed by the legal position on each of the 13 machine categories PHW trains.

No. No UK government body issues a licence to operate plant or a forklift, and no statute requires one. What exists instead is an employer duty: the Health and Safety at Work etc. Act 1974 section 2(2)(c) requires an employer to provide the information, instruction, training and supervision necessary to protect employees, and PUWER 1998 regulation 9 requires that anyone using work equipment has received adequate training in how to use it, the risks involved and the precautions to take.
The Provision and Use of Work Equipment Regulations 1998, regulation 9, requires that anyone using work equipment, including plant and forklifts, has received adequate training for the purposes of health and safety. That covers the methods to be adopted when using the equipment, the risks the equipment may pose, and the precautions to take. The duty to make sure that training has happened sits with the employer.
The Approved Code of Practice L117, Rider-operated lift trucks, sets out how PUWER regulation 9 applies to rider-operated lift trucks, including telescopic handlers, in a three-stage model, and PHW follows the same model for its other plant: basic training in the core skills common to the machine type, job-specific training in the particular attachments, loads and environment the operator will work in, and familiarisation with the specific machine and site the operator will actually use. The employer completes the process with a written authorisation for that operator to use that machine.
Where a machine is used for lifting, for example a forklift raising a load, an excavator placing a pipe, or a lorry loader unloading a delivery, the Lifting Operations and Lifting Equipment Regulations 1998 (LOLER) apply to that lifting function alongside PUWER. LOLER requires the lifting equipment to be thoroughly examined at intervals not exceeding 12 months, or 6 months for lifting accessories and for equipment used to lift people, and requires every lifting operation to be properly planned by a competent person, appropriately supervised and carried out safely. The Approved Code of Practice L113, Safe use of lifting equipment, sets out what that planning and supervision looks like in practice, including for slinging and signalling.
The employer, in writing. Training and assessment give the employer evidence to rely on; the decision to authorise a named operator to use a particular machine, on a particular site, is the employer's own decision and the employer's own responsibility. No certificate makes that decision for them.
They are evidence that a named operator has completed structured training and been formally assessed, in theory and in practice, on a specific machine category. The certificate, the theory and practical assessment records, and the site evidence statement together give an employer something solid to point to when they authorise that operator, and something a site can check. They support the employer's PUWER regulation 9 duty; they do not replace the employer's own decision to authorise.
Each machine category has its own legal position page, covering the training model, where LOLER applies, and the points specific to that machine.
These pages set out the general legal position. They are not legal advice, and a site's own entry requirements are a separate matter from what the law requires.