When arresting someone in the UK, you might be entitled to do so under several different laws. For a 'citizen's arrest', you would be using PACE (the Police and Criminal Evidence Act) of 1981 section 24a. A police officer would use section 24 most of the time, although some other laws (like 'drink driving' laws) might contain separate powers of arrest.

To arrest somebody, an UK police officer would need to have reasonable suspicion of a crime having been committed (whether in the past, present, or a 'reasonable suspicion' of a crime about to be committed), and a reason for the arrest. A valid reason for arrest might be a need for further investigation, a risk of escape, a risk of harm to a person, loss of property / damage to property, harm to a child or vulnerable person, ascertainment of name and address etc.

Upon arrest, an officer would have to read the person they are arresting their 'caution'. This is the UK equivalent of the Miranda Rights.

The caution, in full, is "You do not have to say anything, but it may harm your defence if you do not mention when questioned something which you may later rely on in court. Anything you do say may be given in evidence".