Well.. since this is here

Lets see. WW3 could have been a limited nuclear war, or a first round of nukes were sent, but then they held back the rest, or set an arms treaty to not use the nuclear weapons in the rest of the war.. So probably some places were nuked, say New York, Moscow, N Korea etc. but many areas would be okay, maybe why they were in Montana, well away from any city.
So, Lets take the angle that Cochrane did this himself ( with a small team) in 2063.
He may have been part of some project, or head of a FTL project before the war, with alot of the physics and hardware mapped out, maybe even some prototypes were created. The war happened, all the money and resources dried up, so he decides to do it on his own.
OR.. he wasn't part of any team before, and he was just a genius, and done the physics and math and found that he could build a FTL drive, but needed resources.
Either way of this.. The war happened, he was left to himself, and a small team that he cobbled together to do this..
SO what does he need?
1. A spacecraft, you may be able to do test on Earth, but to see if it really works, you need it in space, so He either asks the government, or whichever govenment is left in Montana to use a leftover ICBM for the ship. Or if the US was that screwed up, he could have just Homesteaded the place..
2. Power source. Antimatter? Maybe but today you need a Very large Particle accelerator to create Micrograms of it, so I'll say no, but a Fusion power source, quite possible. So lets say he buys or builds a fusion reactor to power the ship. (Shuttles use battery power, so you don't need Warp Plasma or Dilthium to power a low yeild Warp drive)
3. Warp ship. They dissasemble the upper half of the Titan missle, and start bulding the ship, now you would need specialized materials, machines to do this. There would be some type of infastructure left to go to to get these things, I don't think he had the help of the governement, because.. there were no government people at the camp, no overseer, no political officer.. nothing. so this was a private project, and it took 10 years to cobble together all the materials, maybe use 3d printers ( theyve been growoing in capability by leaps and bounds) to manufacture the ship.
4. Launch.
Now, is it right now possible for some guy in a garage to do this without government help? Maybe the lab work or protoype, but would need an big investment of capital by a person/coorperation/government to get it going.
But after a Nuclear War? To me its quite possible, he would still need some money to buy stuff, so maybe he had a sponsor in a person, company, or he was himself wealthy, or his team scraped together money, or barter in the Afterwar.