Firstly, obviously a climbing wall is a private business and has the right to do pretty anything reasonable or indeed unreasonable e.g. they can insist that people only attend in yellow clothing on a Tuesday and blue clothing on a Wednesday if they want. Private individuals are then free to argue this is stupid.
In terms of 'lifetime' as stated by the manufacturer, this is obviously a very reasonable thing for people to use for guidance when they are making 'rules' or 'policies' (either for their own guidance, for use in a climbing wall, for use at work etc.). The lifetime is not a statement from the manufacturer that the item becomes dangerous after the lifetime expires; the lifetime only indicates that the product should not become ineffective, unsafe or non-functional due to aging before that time (and is sold on that basis). If an accident occurred due to aging of an item that hadn't otherwise suffered wear/tear/misuse etc. before that lifetime, I imagine a claim could be made against the manufacturer (or more accurately against the retailer who sold the item). After the lifetime expires, I would imagine any such claim would fail.
It is not illegal, even when 'at work', to exceed the manufacturer's stated lifetime of a product. However, in the event of an accident due to the age of the equipment beyond the manufacturer's stated lifetime, I suspect you would be up a very unpleasant waterway without a paddle.
Here is HSE guidance explicitly recommending what sort of things you would need to do, if you were an outdoor centre, to continue using kayaking helmets after their manufacturer's recommended lifetimes (which at only two or three years should make us grateful we get relatively long lifetimes for climbing/caving gear!):
Basically it means you need to take on the due diligence of ensuring the equipment is still safe to use due to age-related issues.
