Assuming the person/people in question have British connections, have you searched Find My Past's "British Nationals Born Overseas 1818-2005"? My recollection is that all records (bmd) for British Nationals registered overseas, were returned to the GRO annually, so should be in their indexes. Embassies, High Commissions and Consulates do not necessarily retain historical records.
The Tanzanian Government will hold very few - if any - records from the Colonial Period, for its own Citizens or for British, other Commonwealth or foreign citizens living in either Tanganyika or Zanzibar at that time. For indigenous people at this time, especially those living in remote villages, birth registrations were simply not necessary or a priority. Informal arrangements were available should a document be needed at a later date e.g. a Tribal elder might declare that he/she remembered the birth of a given child on a (very) approximate date.
Bear in mind that modern Tanzania is one of the 35 poorest countries on earth, so has very limited resources; I fear RITA's website rather oversells what they can possibly deliver, especially to enquirers.
Kevin now in Chester