Hello Swarup!
To grant admission in different Government and Private Nursing Colleges/ Institutions of West Bengal to B.Sc. Nursing courses, Joint Entrance Test ( former JENPAUH now known as JENPAS-UG ) is conducted by West Bengal Joint Entrance Examination Board (WBJEEB) on behalf of Central Selection Committee (Nursing), WB Govt. College of Nursing (SSKM Hospital Campus).
It is followed by an online counseling process for the State quota UG nursing seats.

So,
-Firstly you will have to fill up the online application form available at http://www.wbjeeb.nic.in for JENPAS-UG (2021) i.e. in the next session as for this year, the last date to fill up the application form was 13th February, 2020. Though the examination (JENPAS-UG 2020), which was scheduled to be held on 10th May, has been postponed till further notice due to lockdown all over the country.
*This year, the application fees was Rs. 500/- plus Bank's Service Charges, as applicable.

-After submitting your online application form and paying its fee, you need to download the automatically generated Confirmation page.
In the subsequent days, admit card will be issued and then, you'll finally get to appear in the entrance exam (JENPAS -UG).

-Results will be published in the form of Rank card, which will contain your relevant ranks and total score. Rank/Score lists are never published by the Board.
You will be able to view and download your rank card by logging in to your account with its password.
-A separate detailed notification of counseling/seat allotment and admission will be released in http://www.wbjeeb.in or http://www.wbjeeb.nic.in after publication of result.
-Course and institute wise availability of seats to be provided by Competent Authorities will also be published before counseling and seat allotment.
If you will meet the required cutoff and get selected for Government Nursing College, SSKM Hospital, Kolkata during counseling, you'll get admission here.

You can find the complete details about JENPAS-UG at : http://wbjeeb.in/jenpas-ug
Don't miss out to go once through the Information Bulletin available here carefully before filling up its application form.

You can surely ask your doubts (if any) related to this in the comment section.

I hope this answer will be helpful to you.