Typical timelines by change type
Editing an existing record with a low TTL, such as pointing an A record at a new server, often propagates in a few minutes to an hour. Records with longer TTLs take proportionally longer, up to the TTL value you configured. Adding a brand new record usually appears fast because there was no old answer cached to expire.
Changing your name servers (NS records) is the slow case. That delegation is cached at the top-level domain level, frequently for 24 to 48 hours, so a registrar-level DNS host migration is the one change where the old 'up to two days' warning still genuinely applies.
How to make it faster and verify it
The single most effective trick is to lower the TTL well ahead of a planned change so resolvers hold the old value for a shorter time. Do this a day in advance, make the change, confirm it, then raise the TTL back up. During a name server migration, keep identical records on both the old and new hosts so visitors get a correct answer no matter which one they reach.
Rather than guessing whether you have waited long enough, check the record from many places at once. If nearly every location returns the new value, you are essentially done; if the results are still split, you are inside the TTL window and should wait a little longer.