Am I eligible for UPSC if the board is later declared as 'unrecognised'

The board was in recognised status in 2016 when you passed your 12th. After that only, its recognition was cancelled. So there is no problem in having recognition of your qualification. You can go ahead.
Already you have completed your BBA and MBA from an A+ accreditation University. You got good percentage in both the courses. That itself is an indication that your 12th certificate is valid.
So you can apply for UPSC and all other government posts if you meet the qualifications and other criteria. You will not have any problem and your application will be considered.

As far my knowledge goes you need not worry about your future as you have already completed good education and also secured distinction marks. You create confidence in your self and apply for UPSC exam and at the application stage itself the UPSC would take those which are recognized and would reject which are not recognized as per their records. Some universities or boards are de-recognized for not following certain norms prescribed by the National accreditation agencies every year and that is internal matter to which UPSC may not give that credence. So go ahead and and apply to UPSC and proceed.

Your present worry regarding the validity of your class twelve is disturbing you. During the period of your admission, it enjoyed the recognition status and I hope you might have passed your class twelve in the year 2018 when it was still enjoying the recognition status.
You took admission in BBA on the basis of the validity of class twelve examination.
Hence you need not worry for derecognition of your class twelve.
Since you have added BBA and MBA qualification securing 87 percent and 92 percent respectively enabling you to make eligible for UPSC examination. Even you are eligible to apply all other posts where there is requirement of such qualifications.

Yes, you are eligible to appear for UPSC in your case. You took admission in 2016 and at that time the board was recognised, when you completed your 12th in 2018 then also the board was recognised and that's why you got admission in college and completed BBA and MBA. The recognition of your 12th board was cancelled one year after you passed out. The eligibility criteria for UPSC is a Bachelor's Degree from any university that is incorporated by the Central or State Legislature act in India or from a Deemed University under the UGC ACT. You have completed your BBA and MBA from A+ accredited university so there is no need to worry about that. Academically, you are a good student and I wish you all the best for UPSC.

As per the information provided by you when you took admission in class 11 that is in the year 2016 this board was a recognised one. So you must have passed your class 12 in the year 2018 and still the board was a recognised one and then you moved to some college for your BBA. You have also mentioned that the board was declared unrecognised in the year 2019 so the students who have taken admission in class 11 in 2019 would be affected by that declaration.

So, in your case there is no problem and you can very well go ahead for your plans of applying in UPSC. The concerned verification authorities would find your credential in place as the board was recognised during your clearing of classes 11 and 12.

Q: What are the eligibility criteria in terms with the educational qualifications of a candidate to appear for Union Public Service Commission(UPSC) exam?
A: As per the UPSC rules and guidelines, the minimum education of a candidate to appear for the Union Public Service Commission(UPSC) exam is graduation in any discipline from any recognised Universities incorporated by an Act of the Central or State Legislature in India or other educational institution declared to be deemed University under Section 3 of the University Grants Commission(UGC) Act of 1956.
- There is no minimum percentage required to be eligible.
- Candidate in their final year can also apply by giving proof of the same while applying.
- Only UPSC exam marks are counted, thus marks of your degree have no advantage.

Q: I did my BBA and MBA from an A+ accreditation University and have scored 87% and 92% respectively. I have to appear for the UPSC exam. Will I be eligible for the same?
A: As you have completed your studies in 2016 and during that time your institute or board was recognised, you don't have to worry as anything after your completion should not affect your certification. You have also completed your graduation after your +2 from an A+ accreditation University, so you don't have to worry about recognition of your past board. You stand right to qualify and eligible to appear for the UPSC exam. Do prepare well and go forward to apply for the same.