Huh that's new. I knew you guys don't have any official government ID (and that it was an issue in Blair era) but I didn't knew you don't even have legal names.
We have official government ID (the national insurance number, roughly equivalent to a USian SSN), but names aren't expected to reliably link it in one hop, and we're politically averse to laws that require citizens to produce it in various circumstances, e.g. carrying ID cards, even though in practice most people do carry some form of government-sanctioned ID. Name changing is mostly limited by the fact that it becomes the duty of the person using a different name to laboriously link the new name to any previous names they've used wherever you need some paperwork in a different name. E.g. you need to provide your NI number to be employed, which typically means you need to have a paper trail back to it across any renames you might have had.
Your legal name is whatever you want it to be at any given time, you can just order a new passport or drivers license with ~whatever name you want written on it.
For a passport, the only real requirement is vague evidence that you actually use the name.
You can also have multiple names listed on your passport: for example, if you have a public and private name, or if you're a noble. I've found that the "vague evidence" requirement is actually quite strict, though: even if you use a name in 95% of everyday contexts, the passport office only counts bureaucratic evidence, and there are circular dependencies with most bureaucracies unless you produce a deed poll (which requires you to attest that you only have one name, for some reason: I'm not familiar enough with the law to know whether this is a real requirement, or just ceremonial language like the "I promise that a Catholic will never succeed to the throne" bit in the coronation ceremony). So the de facto rules about passport names are stricter than names in general.
This is gradually getting tightened by the database state for all the usual reasons (hostility to immigrants and trans people); you can get away without ID as long as you don't want to work, claim benefits, rent a house, buy one, or have a bank account. Other than that, you're fine.