I have deleted my old NS records from DigitalOcean, who hosts my website, and I have added new NS records pointing to Cloudflare’s nameservers. It’s been days and I am not seeing my nameserver change in either my Cloudflare dashboard nor in ICANN’s WHOIS lookup.
However, whatsmydns.netdoes reflect that my NS records have changed, and almost every DNS in the world is now showing my NS record as pointing to Cloudflare.
Without knowing the actual domain name, we can’t give a definite answer, but it sounds like you added DNS records for NS, rather than setting the Name Servers at your domain registrar. That would show up at whois.com