Hello Subham,
No need to get confused. It's pretty simple. It is clearly stated in the JEE guidelines that the state from where you give the 12th board, is your state of eligibility, no matter where you live. In your case, since you live in Bihar, but have given 12th board exams from UP Board, then clearly your home state will be Uttar Pradesh. This means that you will be eligible for the 50% reservation in NIT Allahabad but not NIT Patna. Also, for your information, you can submit your caste certificate, if you belong to any category, issued from Bihar.
I hope its clear now!