Do the tld name servers cache the authoritative name servers? For how long?

When I specify the nameserver for a domain in the registrar console I use the nameserver name. Let's say I change the ip of the nameserver. How long would that change take to propagate?

Authoritative nameservers like the TLD servers do not cache data from other sources, they just serve the data that they have. As such, the TLD servers will serve your updates as soon as your changes through the registrar have been processed.
(Traditionally that is, there are authoritative nameserver implementations that can dynamically fetch external data, however, this is not relevant for the TLD server case.)

However, resolver servers all over the internet do cache and their caching behavior will simply be based on the TTL of the relevant records.

That said, if a nameserver name in the delegation requires glue records for bootstrapping purposes, like when the nameserver name is part of the zone that is being delegated (eg example.com. IN NS ns1.example.com.), you also include address records (A/AAAA) together with the NS records in the delegation.
Such glue address records would need to be updated through the registrar in addition to updating the authoritative records in your own zone.