If such an event occurred anytime soon or the near future (highly unlikely IMO), I can only see Bitcoin crash and/or stall in development.
This is assuming that Bitcoin's umbilical cord is to the US dollar (and I think it a fair assumption, no?). As a nation, Bitcoin would only harm US interest in wartime, and drastic measures to severe Bitcoin and communication networks with enemy nations would be supported. Would it be effective? At least somewhat, possibly greatly effective.
In other words, it would be very suspicious and difficult for private citizens to argue their freedom for pursuing, maintaining and developing BTC during wartime - for any nation. Internet & BTC = Global Interest (as a whole). World War = Nation Interest (specific/particular).
I can see Bitcoin being used for small bands of operatives coordinating anonymously online via encrypted messages, paying for military grade weapons via the anonymous Tor gun site and the group being able to amass an army better equipped than some small nations.
So essentially, a small group of people could use Bitcoin to take over a nation.
As far as its effects on a world war. Think the French resistance but being funded completely with Bitcoin.
Yes, but only if Bitcoin's value were stable and applicable across many currencies and cultures. As of now, only private BTC wealthy interests could supply smaller armies with guns, food, and supplies; but it would be those small armies that would have trouble getting guns, food and supplies with Bitcoin, or the private interest would need to deal only with other believers in BTC who could supply. Bitcoin doesn't have any real power right now - regardless how you define 'power' - only potential.
Far future, and if BTC is established, global and independent? Then yes, very effective for funding resistance. And it could transcend any nation's fiat if the internet and personal electronics are seen just as strong and independent.