As said the time frame is debatable and also as said by some, if China is helping in this project, things might be done more quickly then anticipated. Yeah if we work all alone, may be in 20 years we get something, but if we start getting help of someone who itself is making such things already, the timeline is cut drastically.
And as for the cost, why is so hard for you guys to understand simple economics. What the US builds and operates does not means that others will get the same stuff at same cost. Why is China making good quality products at cheaper rates, why we call JF-17 a 4th gen fighter but available at a cost of half of what western option might be available, simple reason, labor is cheap here, in this part of the world. US buys things in dollars and it pays for the operational expenses in dollars, thus its their own currency. It doesn't means we would be getting the nuke sub for 1B$$, it may be for 400m$$ or may be 500m$$ and operational cost could be much lower too.
Do you have the operational expenses of PN subs ?? Even though the buying cost was high, but operational cost would be considerably low compared to may be French ones. And with more and more local production of spares, operational cost comes down.
So end is, its very much doable, but will need time and resources and good economy. And it won't be as expensive as what western options are, heck we are not even knwing what its dimensions and capability are, then how can we say its a 1B$$ sub, may be its much smaller in size and has limited capability compared to what US or other bigger nations operate.
And who knows the news is mistake, may be the navy is talking about AIP conventional subs but with nuke missile capability and someone translated it into a nuke-reactor sub.
My guess is, they are talking about a conventional sub with the capability of carrying nuke missiles, as that is the only platform which can be developed in the 5-7 year time frame.