No need to re-register. Everything is on the TPS page for Nepal.

TPS Continues Through:
As long as the preliminary injunction ordered by the court in Ramos, et al v. Nielsen, et al., No. 18-cv-01554 (N.D. Cal. Oct. 3, 2018) remains in effect, or by other order of the court.

TPS for current beneficiaries under the designation for Nepal will continue, provided that they properly re-registered for TPS during at least one of the following registration periods for their country. If you have not re-registered during at least one of the following registration periods, you should file now and explain your reasons for filing late.

Registration Period for People Who Already Have TPS:

May 22, 2018-July 23, 2018

or

Oct. 26, 2016-Dec. 27, 2016

Employment Authorization Document (EAD) Auto-Extended Through:March 24, 2020

https://www.uscis.gov/humanitarian/temporary-protected-status/temporary-protected-status-designated-country-nepal

Thank you. So we also don't need to file for EAD?

no

But if lets say, I want to re register. Do I need to pay the re-registration fee ($410) for EAD and TPS or its free of cost. I will get new EAD card with validation of March 2020?

I you want to apply for a new ead then you need to pay the application fee for EAD.