I wanted to know more about law exams as I would be pursuing BBA LL.B.

No, you cannot get into a National Law University (NLU) with a ULSAT score. NLU admissions for the 25 participating NLUs is done through the CLAT (Common Law Admission Test). For NLU Delhi the AILET is conducted. ULSAT is for UPES, not NLUs, so you need CLAT or AILET scores.

While UPES itself is the primary institute taking its own ULSAT for its law programs. Other private law colleges in India also consider ULSAT scores alongside other exams like CLAT, with Amity University, Presidency University (Bangalore), Chandigarh University, Parul University, and LPU often listed as institutions accepting ULSAT for their BA. LL.B, B.B.A LL.B, etc.

ULSAT (University of Petroleum & Energy Studies Law School Admission Test) admits students to various UG/PG law programs with a combined seat intake of around 560-590 seats, varying slightly yearly, for courses like B.A. LL.B., B.B.A. LL.B., B.Com. LL.B., B.Tech. LL.B., and LL.M., with specific seats allocated per programme (e.g., B.A. LL.B. often has the highest, around 240). Admissions use ULSAT scores, Board Merit and CLAT.

Yes, you can get admission based on CLAT Score. Apart from that you can also appear for ULSAT Exam conducted by the university. 

ULSAT Exam Result will be published online. Since the whole admission process is completely online nothing will be published offline. Candidates must not trust anything published offline.