For those unaware, urinating in public is usually an offence of breaching of a local by-law, and potentially an offence under s.5 of the Public Order Act 1986 (if it is disorderly behaviour etc). Both can be dealt with by a Penalty Notice for Disorder or, if the circumstances are such, by arresting the offender if the officer believes it necessary to do so.
Can an English police officer compel a private citizen to purchase something?
No. The police have no statutory power to compel someone to purchase water to clean up their mess.
What is their latitude in arresting/fining someone if they refuse to comply?
Officers may, not must, issue a PND or arrest , per 2.4 of PACE Code G
The power of arrest is only exercisable if the constable has reasonable grounds for believing that it is necessary to arrest the person .... and it remains an operational decision at the discretion of the constable to decide ... to issue a penalty notice or take any other action that is open to the officer.
Other such action can include trying sort it matter out by encouraging the offender to see the error of their ways thus avoiding a fine/being arrested.