Most of the tests are PCR or RFLP (restriction fragment length polymorphism) tests. The delay probably comes in transporting a sample, possibly while refrigerated, to a place where these can be done cleanly. Another long delay would be in acquiring reference samples, but I assume the government has already identified those, either from hair samples from their palaces or relatives.
PCR is simple and easy, but it is not the standard for identification. It also is very prone to contamination and is totally uninformative in many cases.
RFLP analysis is harder and requires more DNA and takes longer but is not prone to contamination and can be more informative than PCR. It is Southern blotting based, as Smeghead mentioned above.
A brief rundown of the procedures. For PCR: DNA is very crudely extracted (under 30 minutes probably), and this is often optional depending on the exact PCR. One makes a mixture of enzymes, primers, DNA, and assorted buffers and runs it in a thermocycler for usually 2-6 hours. It is then run on a gel for another few hours to overnight. This is the most rapid.
For RFLP, DNA is finely extracted (6 hours or more of protocol). It is then digested with several restriction enzymes (2 hours to overnight). It is then run on a gel slowly (usually overnight). The gel is then washed and denatured for 3 hours or so, and then transferred onto a nylon or nitrocellulose membrane for at least 4 hours (also can go overnight). The next morning, the membrane is washed well. Concurrently, you prepare a radioactive probe by labeling another piece of DNA. This is hybridized, usually overnight. The next day it is washed well and set up with radioactive film. Depending on the strength of the probe (length and sequence and freshness of isotope used for labeling), this incubation lasts anywhere from 30 minutes to 2 weeks.
Of course this process can be streamlined, but the above is how I do it. Count on 4 days, at least, as Smeghead said. First day extract DNA, cut DNA, and run the gel overnight. Second day, wash the gel and transfer it. Third day, label probe, hybridize with the membrane. Fourth day, wash and set it up with film.