In the UK: the First World War was a different time when both police and firefighting organisations were in their infancy, with everything being small, local and greatly relying on the community effort.

The draft did lead to severe lack of staff and "Special Constables" (volunteers) were largely used instead of regular police. The Special Constables were established much earlier but in the past their presence required "tumult, riot, or felony". In 1914 "The Special Constabulary Act" updated their duties "during the present war". That limitation was in turn abolished in 1923.

There was a "Women Police Volunteers" service established in 1914. They wore full uniform and would patrol the streets but had limited scope, concerning themselves with specific activities ( to help with harassment and abuse, or women being recruited for prostitution). In times of war every little help is quite essential as it allows for more police to be available elsewhere. The women police enforced the "Defence of the Realm" act, public decency, and the supervision of female workers such as the ones employed in munition factories. In 1915 a member of the Women Police Service became the first female police constable with full power of arrest. From 1919 onward women were officially accepted into the regular police forces, although at the start of WWII they numbered only 300.

The air raids led to the creation of a whole range of "Air Raid Precautions", including the "Raid Wardens Service", set up in 1937. That meant every council had ambulance drivers and rescue parties, and liaisons with police and fire brigades. In 1941 the ARP became the "Civil Defence Service", which took care of fire, rescue, first aid and stretcher parties, and gas decontamination - they were the "first responders", with a total number of 1.9 million.

The mistake of drafting essential professionals was fixed in 1938 when the Schedule of Reserved Occupations was drawn. Full-time police and firemen were now exempt from being called to war, just like many other people whose manufacturing or service jobs were vital to the everyday functioning of the country. There was a long list of such "reserved occupations", the preliminary version of which can be seen here:

http://anguline.co.uk/Free/Reserved.pdf

There was also an Auxilliary Fire Service created 1938 which complemented the main one but its members were not professionals (and not particularly skilled as firefighters). They did join the army and during the war the AFS consisted of those too old or too young to go to be conscripted. In 1941 it became part of the National Fire Service.

"Women's Voluntary Service for Civil Defence" also appeared in 1938. Their job was set up to "make known to every household what it can do to protect itself and the community” but during the raids they did a lot of hands-on work: helping with evacuation, setting up canteens, and running Incident Inquiry Points were people could find out about loved ones in the bombed areas. Without the women defence volunteers all that would have been extra work for the fire brigade and the police.

The war was the first time women joined the professional fire brigades. They did fight fires and there were women pump crews, such as the one in Rochester in Kent, or the all-woman fire station in Northumberland where there were no men at all to do the work. Although their main tasks were to drive the crew or to be a fire watcher, which was essential with all the incendiary bombs coming down. Major buildings like the Canterbury Cathedral had constant fire watch which prevented the whole building being set on fire despite several direct hits on the roof. During the war a separate rank system for women was developed to recognise their bravery. Some got awards, such as 21-year-old Gillian Tanner who received the George medal for bravery, delivering petrol to fire pumps while the London docks were bombed in 1940. And yet in 1943 the government rejected the Fire Brigades Union demand for equal pay for women, claiming that equal pay for equal work was not generally accepted. Women still got only 75% of a man's salary.

The British war effort was assisted also by volunteer firefighters, some of which even helped with the evacuation of Dunkirk onboard "Masey Shaw", a boat named after the first Chief Officer of the Metropolitan Fire Brigade. That boat alone evacuated over 500 troops and even the flag was used to bandage a soldier's arm.

In 1939 "War Reserve Police" appeared as well, made up of volunteers with the full powers of a police officer plus extra duties: enforcing blackouts, assisting evacuations, combating the black market and even capturing deserters. They would wear metal Brodie helmets instead of the regular police hats and were armed with a rifle so they could assist the armed forces, including against enemy sabotage.

In 1939 some 3000 reservists (people who had previously served in the military) left the police and returned to their units. During the war 16,500 more followed. With 60,000 policemen before the war, by 1944 there were 43,000 regular police officers, 17,000 WRSs and Special Constables and 385 women police. Older men would often remain in work instead of retiring or enlisted as reserve constables or specials.

Further reads: "Firemen at War" by N.Wallington, "ARP and Civil Defence in the Second World War" by P.Doyle, "Forged in Fire: History of the Fire Brigades Union" by V.Bailey (former union official).

Fire departments in the modern sense really only started emerging in the 19th century so a lot of smaller towns didn't have one or relied on volunteers. However, most major European cities died have fire departments at this point and many towns did as well.

Most countries considered fire fighting and medical personnel as vital especially given the fears of strategic bombing in the inter-war period and they were less likely to be drafted into combat units and if they were drafted they were usually drafted to be military fire-fighters or medics.

So most countries didn't run into a large manpower shortage in firefighters because they didn't have anything close to universal conscription and fire-fighters especially older ones were not first on the list to get drafted. For example, in the the US only a fifth of men between the ages of 18 and 45 were actually drafted (The National WW2 museum) and the first to be drafted tended to be younger, the unemployed, and those working in industries that weren't considered to be as vital so emergency personnel were not usually drafted and many fire departments were volunteers anyway and were somewhat less common then they are today.

The UK specifically created a service known as the auxiliary fire service. Professional firefighters in large cities were not drafted. They actually needed more firefighters to combat the bombing so instead men that were too young or old for the draft could volunteer to be part of the auxiliary fire service Women made up a large portion of the fire spotters and drivers but were not usually called upon to fight the fires themselves.(Source:London fire department.gov.uk) Bucket brigades also would sometimes assist the professional and auxiliary fire fighters.

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In Germany, an organization known as the Feuerschutzpolizei functioned in much the same ways as an auxiliary of older and younger men, along with some conscripted Eastern Europeans supplemented the undrafted city fire departments. In the case of firestorms that overwhelmed these fire departments. Everyone and anyone especially off-duty military personnel were organized by the police, SS, or military officers into ad-hoc fire and rescue teams. These were known as the Pflichtfeuerwehr or compulsory fire brigades. When the fire department was overwhelmed it was part of citizens responsibility to try and save their city. They would also sometimes use POWs or impressed workers. I haven't heard of concentration camp victims having been used in this role so I can't speak to that.