The answer to 'how to unclog emergency plumbing' is to first try using a plunger. Firmly plunge the clogged drain up and down 10-15 times to dislodge the blockage. If that doesn't work, you can try using a drain snake or auger to physically break up the clog. Carefully insert the snake down the drain and turn the handle to extend the cable, pushing through the blockage.
If the clog persists, you may need to disassemble the drain pipes under the sink to access and remove the blockage manually. This is a more involved process, so calling a professional plumber is recommended for this step. They have the right tools and expertise to safely clear the clog without causing further damage.