Hello Aspirant,
No there will not be any problem, but you have to just fill Jharkhand board as your class 10th passing board and West Bengal board as class 12th appearing/passing board in NEET application form.
As NEET 2021 applications are not out yet so after it gets released you can apply for NEET 2021 by following the given process :-
> Very first step is to visit https://ntaneet.nic.in/ntaneet/welcome.aspx and fill out the application form, then on first menu you have to do registration using your basic details such as name, father's name, mother's name, contact details and so on.
> Now on second menu item you will have to fill details like your academic performance, address, ID type, medium of question paper and so on.
> Then on Next menu item you have to upload some images such as your passport size photograph (10-200 Kb in JPG format), Signature (4-30 Kb in JPG Format), Class 10th certificate (100-300 Kb in JPG Format), Left-Hand Thumb impression (10-50 Kb in JPG format) and postcard size photograph (50-300 Kb in JPG Format).
> Then you have to proceed for payment of application fee which can be done only in online mode either through Credit/Debit card or UPI or NEFT/RTGS.
> Then after you have to take the print-out of the NEET Acknowledgement receipt.
To get live updates and complete details about NEET application kindly visit the link given below :-
https://medicine.careers360.com/articles/neet-application-form
I hope this information helps you.
Good Luck!!
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