Moving the domain to another hosting provider does not require your domain name to be transferred, and it doesn’t even require the old hosting provider to approve anything.
All you need to do is to edit the DNS settings of your domain to point the traffic from your old hosting provider to the new one, which is typically done at place where you bought your domain name.
The only situation I can think of where a release of a domain name would need to be authorized is if you’re transferring the domain name registration itself. Some (but not all) domain extensions require this approval. How to approve the transfer varies by domain provider, and not all domain providers let you approve transfers.
However, this doesn’t mean you have to wait. Even while the domain is being transferred, you can update the DNS settings of your domain at your old domain provider to point the domain to FastHosts, so you can start using it while the transfer is still in progress. For exact instructions on what to set up where, please contact FastHosts.
In any case, we cannot help you with this, because we’re not your domain name provider.