NO - there is no law that we are aware of that states you MUST get appliances "PAT TESTED".
However, there are numerous laws / legislation that you must comply with to ensure electrical items are safe - it is up to you HOW you comply, but PAT testing is the usual method.

To quote the Health And Safety Executive

The Electricity at Work Regulations 1989 require that any electrical equipment that has the potential to cause injury is maintained in a safe condition. However, the Regulations do not specify what needs to be done, by whom or how frequently (ie they don't make inspection or testing of electrical appliances a legal requirement, nor do they make it a legal requirement to undertake this annually).
Again, it is important to stress this point, the law does NOT specifically state that pat testing must occur, what it does state is you have legal obligations to ensure safety.

The HSE states (in regards to is pat testing compulsory)

No. The law simply requires an employer to ensure that their electrical equipment is maintained in order to prevent danger. It does not say how this should be done or how often. Employers should take a risk-based approach, considering the type of equipment and what it is being used for. If it is used regularly and moved a lot e.g. a floor cleaner or a kettle, testing (along with visual checks) can be an important part of an effective maintenance regime giving employers confidence that they are doing what is necessary to help them meet their legal duties. HSE provides guidance on how to maintain equipment including the use of PAT.
So, if PAT testing is not the 'law' why should you pay to get it done?

The answer is simple - how else are you going to show that you are making reasonable efforts to comply with the laws and legislation?

You should be complying with all the following regulation (and possibly more, depending on what you do).

Health and Safety at Work Act of 1974
Electricity at Work Regulations of 1989
Provision and Use of Work Equipment regulations of 1998
Management of Health and Safety at Work regulations of 1999.

If something happens, an employee or member of the public receives an electric shock, or a fire is started by an electrical appliance then questions WILL be asked by the authorities, insurance companies, investigators etc.
If you have no scheme in place to maintain or check your appliances then how will you be able to show you fulfilled your legal duties, that you observed your legal duties of care?

You may well have to answer these questions in a court of law - duty holders are responsible and ultimately the buck stops with them when something goes badly wrong.

Also think of the business costs if an employee is injured, or your business has to cease trading due to a fire.