The Transfer Notification is sent from your Provincial CFO's office, once they have approved you (yet again) to be in possession of certain fire-arms. If you need additional paper-work like an up-dated ATT, they will usually include it in the same envelope.
Registration, however, is handled by the CFC in Miramichi. They will print new Registration Certificates in batches to save work (once per week), then they all go in the mail. No further approval is necessary, but this will delay your Certificate by the time until they print, plus mail delivery time from New Brunswick.
Read your ATT. It will most likely state the condition that you must be in ‘possession’ of the Certificate, meaning you should have it on your person, rather than being the ‘holder’ of it, meaning that it has legally been issued to you and not revoked (and a computer check can verify your holder status).
Is it possible that a police officer will accept the Transfer Notification (with Certificate number) as proof of compliance if it ever comes to that? Maybe, but that's not how the regulation is worded. Is it probable that no-one will ever even know you were treading on this thin ice? Absolutely. But technically, you must wait until you have the Certificate for each fire-arm you will be transporting. In the mean-time, try calling your CFO and requesting a Short-Term ATT, perhaps for participation in a target competition, without this condition attached; it seems reasonable, so why would they refuse?
Jump to the Legalese forum here to get the lawyers involved in deciphering all the complicated language of the Firearms Act.