Hii Adibh!
You can definitely go for attempting neet 2020 while you are in your first year of biochemistry course. Although, you must check the eligibility criteria first. There's no rule as such that you cannot attempt neet once you take admissions in some other course. But, when you get admission through neet in BSc nursing, you ought to leave your present subjects and produce a leaving certificate from your current college in order to get admission in new College. You cannot pursue 2 courses simultaneously and it's not at all possible to do so. When you are confirmed of getting admission in BSc nursing, apply for the leaving certificate in your present college. Hope this helps.
Good luck!