If you want to know what the politicians will do, consult your
Magic 8-Ball.
Meanwhile, Treasury has been through this fire drill before, so they've been stockpiling the incoming cash flow and taking a number of other extraordinary measures to meet the government's existing obligations without exceeding the debt limit. Since Treasury isn't authorized to issue new debt, once funds raised by these extraordinary actions are exhausted, they will be forced to delay payments for government services, and if the delays go on long enough, they could be forced to default on legal debt obligations.
They will try really hard to avoid defaulting on the legal debt obligations, the bonds and notes that come due, and the coupon interest to be paid. Any technical default, or failure to pay on time, will cause a drop in the value of US debt and effectively raise the cost of any future borrowing.
Here's a
GAO report on the debt management problem.