Gold will probably still be valuable if society and its infrastructure collapses. What's Bitcoin without the internet? Should something like WW3 occur, can we really assume that there'll still be a unified internet and that it'll be working well enough to allow for bitcoin to work?

One could argue that you could use other ways to share the blockchain (radio waves or even sharing physical storage) but how do you settle transactions that way? You'd have to broadcast your transaction somehow, wait for a miner to pick it up, mine it and then broadcast it to the rest of the network and wait for other nodes to see and accept the new block and then build on top of it. Sounds like a nightmare over high latency networks, you could wait for days until you're sure that your transaction is accepted.

What if a certain region is cut off for a while (say, a country)? They'll make transactions, mine them and then when they're reconnected to the rest of the world their block "branch" is immediately overwritten by the outside chain because it had a greater hash rate. Actually for most countries that wouldn't even happen that way because they probably won't have the hash power required to mine a single block in weeks, effectively making Bitcoin unusable.

I don't think Bitcoin would be a very practical currency for Mad Max. Better invest in bottle caps.