The UK doesn’t have any specific legislation aimed at theme park rides. In fact, the UK doesn’t have a lot of specific legislation, which is something that the Grenfell Tower fire inquiry has highlighted. However, there is general safety legislation that can be applied to theme park rides. Most notably the Health and Safety at Work Act. When theme parks are prosecuted for breaches of health and safety, it’s often for not complying with the Health and Safety at Work Act, which gives businesses a duty to manage risk.
In the UK we do have something called Approved Codes of Practice (ACOPS) which are written by the HSE (Health and Safety Executive) in conjunction with industries. Following the ACOPS is supposed to demonstrate that a business is following the Health and Safety at Work Act, although the ACOPS aren’t technically legislation. There are lots of ACOPS that would apply to theme park rides, but the one aimed specifically at theme park rides is HGS175.
https://www.hse.gov.uk/pubns/books/hsg175.htm
HSG175 says that an amusement ride should have a safety doc (there are a minority of rides such as go karts that may be inspected under other schemes). Until recently all safety docs were issued under a scheme called ADIPS, although there is an alternative scheme now called LEAPS which some parks use, including Paultons Park and Flamingo Land. Blackpool Pleasure Beach currently uses ADIPS.
https://adips.co.uk/
The Leisure Equipment & Asset Protection Scheme (LEAPS) ensures that the engineering used day in and day out for fun and play is safe and reliable.
www.leapscheme.com
The inspectors who issued the safety docs may require a park to modify a ride (for example adding seatbelts) for them to issue a new safety doc. This is generally to make sure the ride complies with international standards (EN standards and ISO standards in the UK).
Insurance companies don’t normally get that involved in how theme parks operate, but they probably would require all the rides to have a safety doc, so if the independent inspector from the inspection body required them to add seatbelts, the insurance company effectively would do so by proxy.
The legal situation around theme park rides is complex with criminal and civil law, sentencing guidelines, precedents, ACOPS and technical standards, and various pieces of legislation that have been bolted onto the Health and Safety at Work Act, often in response to EU directives from when we were in the EU. But there’s no legislation specifically around theme park ride design.
Do legacy rides need modifying to bring them up to current standards, or are they ‘grandfathered’? It’s a complicated area.