Let me end by asking if there are any scientific types on here. Why does DNA testing take weeks or longer? What all is involved?
Once you have the DNA sample isolated, the testing itself is reasonably rapid. The PCR/DNA sequencing (depending on the type of test) reactions and separation of fragments for display and analysis doesn't really take that long. They have reference DNA samples (from YY's dad, mom, and maybe even from YY herself if they were able to get DNA from her hair brush, toothbrush etc..) to compare the DNA from the remains to, so it's just a matter of running the reactions, displaying the results and seeing if they match. If they don't match YY's reference samples, then they can compare the results from the remains to a database of DNA test results from known missing persons to see if there is a match. Doing all that would be just a matter of days.
The big sticking point is in isolating usable DNA from the remains. This is what can delay things. If you get a relatively fresh body, it's easy to get usable DNA from the soft tissue. However, the longer things decay, it becomes harder and harder to isolate DNA. The techniques become more lengthy and involved. They may have to resort to doing rounds of PCR amplification to get useable quantities of genomic DNA for testing -and that introduces another problem: they have to try to ensure they have amplified the subject DNA, and not amplified some DNA that came from contamination. That's one of the reasons you see them wearing hazmat suits a lot of times when they are collecting things; it's not so much that the collectors are being protected from the environment they are working in. Instead, they are trying as best they can to keep from contaminating the samples they collect with their own DNA....
Bones present an additional problem. Soft tissues are typically very cellular (genomic DNA is contained in the nucleus of the cell, so the more cells a tissue has, the more DNA there will be. Genomic DNA is usually what is used for most routine DNA testing). Bone tissue contains few cells, and they are embedded in a calcified matrix. So, you are often faced with three problems when trying to get usable DNA from bones: you have relatively few cells to start with, the procedures for extracting it from that hard, calcified matrix are more involved, and since skeletal remains have usually been decaying for a long time, much of the DNA will be degraded and gone to begin with. Thus, you will have to do amplification probably, and worry about contamination. This all adds up to the ID process usually taking longer. Often times with bones, they will try to do something called mitochondrial DNA testing -because you will often be able to get more of this kind of DNA. Mitochondrial DNA is totally inherited from your mother -so you can compare the mitochondrial DNA sequence from remains to a missing person's mother to confirm if it is them or not. I am assuming that YY's mother gave a DNA sample for reference sample purposes, so they may do this type of test if necessary.
One thing is for sure. If there were other items found with the body and they are doing testing for BC's DNA, the length of time it takes to do this testing will play no role in how long it takes for them to announce who this is. Once they ID the person and notify the family, they will make public who it is. Any positive ID of BC's DNA on material found there would probably only be made public when presented at the trial.