Assuming you’ve website.tld you just have to create two “child name servers”* eg. ns1.website.tld + ns2.website.tld and set their respective “glue records” (IP addresses). Your register needs to be able to create and publish those to the zone above for it to work. Not sure if that’s the case with yours but it seems to be possible.
The term “child name servers” is used by some providers to define those kinds of records and it may change from provider to provider.
I don’t understand how this can work because ns1.website.tld would be served by my dns server which is not yet known by others.
That’s because they aren’t served by your DNS server. Remember the “publish those to the zone above for it to work”? What happens is that your domain registrar has to publish your glue record to the TLD zone.
If you run dig +trace +additional google.com SOA you’ll see:
Ding asking a root dns server (xyz.gtld-servers.net) who’s the name server for google.com
Root server will provide you with NS record naming ns4.google.com.
… and also return A record for that name, 216.239.38.10. That’s the “additional” response that serves the glue record.
Then dig will proceed to call 216.239.38.10 and ask what’s the record for google.com. That’s how DNS and glue records work and also why it isn’t a circular dependency like you were thinking it was.