What can I study after class 12th from science stream with maths?

No there should not be any hassle in case of admission after a gap year in private universities. Every year many students take gap year or some like to call it a drop year to take their time to reflect and decide on their eventual career path

Yes, you can apply for BTech course at the [Chettinad College of Engineering and Technology (CCET) Karur] before your 12th results are announced. Preliminary admission applications are often accepted for students awaiting their Class 12 results, though final admission requires passing the exam.

Yes it is possible to apply for Nursing without PCB in Class 12 but your options are limited to Diploma courses like ANM course.

But you need to clear Class 12 for BSc Nursing.

No, a gap year after Class 12th won't affect your admission to any private Nursing college in Uttar Pradesh. The only requirement for B.Sc. Nursing requires a candidate to have passed Class 12 from any recognised board with a minimum mark of 45% in PCB together and passed in English individually.