There is a power of arrest under s. 24 A of the Police and Criminal Evidence Act 1984:
(1)A person other than a constable may arrest without a warrant—
(a)anyone who is in the act of committing an indictable offence;
(b)anyone whom he has reasonable grounds for suspecting to be committing an indictable offence.
(2)Where an indictable offence has been committed, a person other than a constable may arrest without a warrant—
(a)anyone who is guilty of the offence;
(b)anyone whom he has reasonable grounds for suspecting to be guilty of it.
(3)But the power of summary arrest conferred by subsection (1) or (2) is exercisable only if—
(a)the person making the arrest has reasonable grounds for believing that for any of the reasons mentioned in subsection (4) it is necessary to arrest the person in question; and
(b)it appears to the person making the arrest that it is not reasonably practicable for a constable to make it instead.
(4)The reasons are to prevent the person in question—
(a)causing physical injury to himself or any other person;
(b)suffering physical injury;
(c)causing loss of or damage to property; or
(d)making off before a constable can assume responsibility for him.
(5)This section does not apply in relation to an offence under Part 3 or 3A of the Public Order Act 1986
So for it to be lawful, you have to know that the offence is indictable, have reasonable grounds for believing that one of the reasons listed applies, and for it not to be reasonably practicable for a police officer to make the arrest.
In practice therefore, it's very rare. And basically inadvisable.