[Legbreaker]

Here in Australia I think in regional centres not a lot would change, although fuel is likely to be limited and horse patrols probably become more common than vehicle borne. Fuel would be reserved for emergencies.
The cities are on the other hand likely to suffer badly from nukes, particularly Sydney, Melbourne, Newcastle, Wollongong, and perhaps Adelaide, Darwin, Brisbane, and Perth.
Army reserve units as well as 1st Armoured Regiment would be tasked with assisting the Police and other emergency services in and around the cities. Again, with fuel having to be rationed (nukes having taken out the refineries to deny NATO, particularly those western forces in Korea), much of the day to day work would be done on foot.
Civilian ownership and use of oil based fuels and lubricates would be strictly rationed to those who can prove a genuine need (farmers, food transportation, medical staff, and the like). Martial law may need to be implemented in some areas (particularly those with military units assisting the police, or around vital installations and facilities).

After a few months, perhaps a year, civilians may be "drafted" into the police, and receive a brief training course. These draftees would probably be used for administrative tasks, guarding of prisoners, and other relatively simple jobs, freeing up the fully trained police for the streets. Likewise, there's likely to be a influx of volunteers to the military, perhaps just so they can guarantee being fed each day, or in the hopes of being sent to Korea (the only conflict zone outside Australia and New Guinea) to give the Ruskies a little payback.

It's unlikely we'd see any deputisations here - it's a completely different culture when it comes to law enforcement to the US. Only those who have received training would be allowed to act as police and carry a weapon, although everyone would be encouraged to assist where possible with information on criminal activities.