Absolutely nothing will change.  The debt limit is raised every few years with the party in power claiming cats and dogs and assorted ocelots will rain down if it's not lifted, while the party without power claims the other is totally reckless and irresponsible.  A few years later the two parties swap positions and go at it again.  No one is dumb enough on either side of the political divide to risk default over a few political sticking points.

If both political parties suffer from a bout of insanity and refuse to agree on the new debt limit, bitcoin's value will plummet.  It's a pure speculative commodity (not currency) that has no impact upon 99.999% of the world's population.  Ask a villager in India whether they would prefer a bitcoin or gram of gold.  Ask a Chinese businessman the same question.  I bet a Greek factory worker would give the very same answer.