nameservers

I need to set external name servers for my subdomain. Do I also need to set A record for that subdomain? Why after adding the NS records they still did not reflect at Nameserver Check tool? How long does it take for the records to be cached by another name servers?

I have been a member of zoneedit like maybe 12 years or more. I had been a member for 5 years before someone bought the business. So, why is it so hard to find the name server information? That information should be the very first thing anyone should see when they use the free service...

I've potentially got myself in a bit of a pickle. Long story short, I need to setup new DNS for a charity/NFP where they have a specific sub-domain that has to be delegated to different, third-party nameservers (the third party manages their CRM and has a discrete MX for the sub-domain etc...