It looks like around 7-8 weeks ship time. See update below. (Inserted new info at top, ahead of the background info on the avisos ships and routes.)
--- UPDATE ---
Finally found actual info on the travel time for the avisos. This book concerning the HMS Centurion actually lists the transit time for the avisos San Lorenzo as 41 days. This put the ship at Puerto Rico. Leaving there on Jan 31st, detained by British on Feb 3rd, and after rescue and various engagements reaches Santiago Cuba on Feb 12, after 12 more days. Then it apparently took two more weeks (inland Travel?) before the message was finally delivered to the Governor of Cuba, Juan de Prado,on Feb 26th 1762, informing him that Spain and Britain were at war. So 53 days travel time to Cuba, plus a couple more weeks delivery time.
(And this just got us to Havana.)
Now, some background about the mail service.First of all, the correspondence between Spain and Mexico was carried by specialized vessels called avisos:
The communication between the Spanish government and the Viceroys in
America was carried out by small and fast vessels called avisos that
operated under the directions of the Secretariship of State.
Initially, these ships were intended to announce in America and Spain
the departure of the Flota, and the Galleons, respectively. However,
they ended up carrying with them all the colonial correspondences. On
20 July 1718, the Crown established that an annual number of eight
avisos would have to be employed to carry the colonial correspondence.
Two years later, on 31 May 1720, the Consulado de Cadiz and the Crown
signed an agreement.22 The Consulado agreed to add another eight
vessels to work as avisos every year. By the 1730s, the avisos arrived
at Buenos Aires, Cartagena de Indias and Havana on a monthly basis.
from Mobilizing resources for war: the British and Spanish
Later, we see that the use of these packet-boats is reconfirmed:
Charles III in 1764 established monthly mail packet boats between
Corunna and Havana and these were permitted to transport goods to the
extent of half their cargo Every two months a similar packet boat went
to Buenos Ayres and there were American post routes connected with it
In 1765 came the great advance.
from: The Spanish Colonial
By Wilhelm Roscher
We can also learn a little more concerning the avisos because a couple of them have been found off the coast of Florida:
Ships thought to have wrecked off the Jupiter coast during the 1600s
include the San Miguel Archangel and the San Francisco y San Antonio.
Both ships were avisos, Spanish courier ships weighing 60 tons or
less. The avisos were well-armed, but speed was their best defense,
which allowed them to outrun larger, better-armed vessels. Avisos were
primarily used to transport correspondence to and from the Spanish
throne, but were known to carry goods and valuables.
Now we can compare this size to that of the ships most often thought of when we think Spanish fleet, the Galleon
...while galleons were mostly under 500 tons, although the Manila
galleons were to reach up to 2,000 tons.
I did find a little more describing the actual route generally followed by the trade fleet:
The ships departed from Seville (later from Cadiz) and sailed down the
coast of Africa to the Canary Islands, where they stopped for
supplies. They then turned west to take advantage of the trade winds
and, after sailing about a month or more, entered the Caribbean
southeast of Puerto Rico. Here the convoy split into two fleets: the
Tierra Firme (Spanish name for the South American mainland) and the
The New Spain fleet sailed on to the port of Veracruz in New Spain
(present day Mexico).
(and for the return voyage)
The two fleets met up in Havana and made preparations for the return
trip to Spain. When they left Havana, the combined fleet sailed along
the east coast of Florida and rode the Gulf Stream—a strong, warm
ocean current that originates in the Gulf of Mexico and flows north
along the east coast of the United States— north as far as Cape
Canaveral before heading east towards Spain.
above from this National Park