What would happen if you were arrested, had no form of ID at all on you, had not previously been in trouble so no fingerprint records were filed on you, and refused to give your identity. What could they do? Keep you until you relented? Follow you home?
Depends what the offence is but if serious enough to be arrestable, they would deal with you in the usual way. Then put you before the court either to be dealt with or to be remanded in custody. You obviously wouldn't get bail.
You could be charged with wasting police time, as they will have to spend time trying to identify you.
If you refuse to give your details then you are giving the police a reason to arrest you, and they can hold you for as long as it takes to establish your details. You will not get bail.
This is under s.25 PACE.
Everybody arrested would simply refuse to give their details if it meant they'd be sent home without charge.