I don't own or watch NOW on a PC, but is there any error message or error code when you try ?
Also I am assuming a different internet browser doesn't help and you are not using a VPN and are watching in the UK if you are a UK NOW customer.
What is the make and model of your new laptop, and what version of Windows is it running?
I own a laptop on which I can successfully run Now, and the steps I take are:-
Open my choice of browser (I like Firefox, but any good browser should work).
Type NowTV.com into the search bar. The Now screen should appear with details of programmes.
Choose Sign In, and give your Now user Id and password.
You should then be able to choose any programme or live channel that is covered by a membership you have, and watch it.
What, exactly, goes wrong for you where, in this process?
The last step. Inspite of having monthly Sports Membership and Boost Membership ( Free Trail), when opened in a browser on a PC, it doesn't stream the Live Sport! What's the point of opting for membership if it doesn't work on a browser! No error message. It just doesn't show the live sport
So you navigate to Sport, choose it, see all the items listed under Watch Live, click on the forward arrow on any one of them, and - what happens?