If you have tried to join a public server after purchasing items in a private server within the first few minutes of an update, this is likely due to the update having not been released to public servers yet.
When the Adopt Me Team releases a new update, we typically release the update first on private servers. Usually within a few minutes these updates will then be pushed to all public servers, however, circumstances can occasionally occur where a release is delayed. If there is a significant delay, we will aim to inform you on our social channels and here on our Helpdesk.
Rest assured, once the update is available on public servers, any items you purchased in private servers will be available in all servers you play the game on.