No. Not gonna happen no matter what sabre rattling and macho name calling some leaders engage in.
If the five permanent member states of the UN Security Council did not possess a significant nuclear weapon deterrent, then yes, a conventionally armed WWIII may be possible. But it would be fairly short lived and will be fought on the Eurasian land mass (Eastern Europe) South China seas or the Middle East. Think Gulf Wars scale and WWII Pacific navy battles. I don't agree with Trinculo on the point of conscription. There would be plenty of people willing to volunteer and no-one that were not forced to, would contemplate surrender.
Modern warfare relies on state-of-the-art weaponry, which is neither easy to produce nor is it cheap. (Missile destroyer or attack submarine circa £1bn+ each, Fighter aircraft circa £100m each etc.) It requires an extremely highly skilled engineering workforce and highly trained operational personnel which takes years to hone. They cannot be mass produced anywhere near a very small fraction of the scale of WWII. They require the combined production resources of many countries around the globe.
In a war, if supply capability gets cut off, the production chain collapses, no more high-tech weapons produced. Cruise missiles would go after key production and storage facilities, as demonstrated last Saturday in Syria. Transport on the high seas is by enormous and slow container ships. These would be ridiculously easy targets for modern nuclear powered hunter-killer submarines. Deplete weapon stocks and cut ability to transport them - no more war.
Of course, once the high technology weapons are depleted, a conventionally armed military from a country with a huge quantity of legacy assets and sufficient conscript manpower, may prevail. However, unless said country or alliance has a significant expeditionary capability to execute sustained campaigns thousands of miles from their home bases, then they cannot project power and will fail.
The only country capable of doing that at present is the U.S.
The U.K. can project limited regional power in the Atlantic, Mediterranean and North Sea areas. This is because the U.K has global bases in these areas and has a potent navy to support operations.
Even China and Russia does not really have the same capability to sustain power projection far outside of their immediate land borders.
Budding powers like India are geared for local defence only.
But those nuclear deterrents are the weapons that have stopped a regional conventionally armed war from escalating to WWIII. That would require one nuclear armed state with long range ballistic missile delivery systems, to launch a pre-emptive first strike against another or against an ally of a nuclear armed state. Even before those missiles hit their targets, retaliatory missiles would already be on their way.
Only then it would be sayonara and thanks for the fish.