John Cabot doesn't get the idea to mount an Atlantic expedition, and instead focuses on the financing and managing of construction projects for various cities, as he did in Valencia and Seville in OTL. He still wants to put a lot of distance between him and his Venetian debtors, so Cabot still moves to England. I'm thinking from 1495 on he makes a living by improving harbors, building bridges, walls, and maybe even ships in English cities from Bristol to London. Tales of Portuguese discovery of islands in the South Atlantic do excite his imagination, but he's getting old and already has an established business in England, so he forgoes any plans for adventure across the sea. He dies in 1506.
Let's stick with this story for Cabot. The Basque, English, and assorted other cod fishermen may make landfall in North America much earlier than OTL as LSCatilina argues, but as of now I do not see a reason why Cabot would be involved in exploration or why those fishermen's voyage would have occurred so soon after the POD. The 1540's sounds like a more reasonable date to me.
So then, we have the Portuguese discovery of a southern continent in 1503, prompting the Spanish to organize a convention in 1505 at which it is decided that in 1507 Rodrigo de Bastidas will lead an expedition consisting of five carracks west in hopes of catching up with the Portuguese. On board the expedition are Vasco Nunez de Balboa and Diego Velazquez de Cuellar, veterans of the Italian Wars.
As to a more northerly route, I'm not sure. Remember, the discoveries of the Americas (for which we will need to come up with a new name) occurred even farther south than OTL. At this point, do the Spanish have any proof of new land on the same latitude as Iberia outside of the tails of some cod fishermen?
A note on the Aztec discussion: the discovery of the Americas occurs in TTL 11 years later than in OTL. Of course that completely changes the course of events, but to give us a sense of time, if European exploration of Mesoamerica is pushed back 11 years than the Aztecs would be left alone until 1528. Montezuma would have been 62 years old. I don't know how long on average Aztecs lived, but it seems to me like he might die of old age soon either before or after contact. As Cuāuhtemōc points out, there were multiple claimants to the thrown. Could we see an Aztec war of succession, or do you Amerindian experts think the succession would have gone without a hitch?
Here is a list of other characters
-Juan de Grijalva; 18 at the time of the 1507 expedition
-Francisco de Montejo; 28 at the time of the 1507 expedition
-Pedro de Alvaroda; 12 or 22 at the time of the 1507 expedition
-Juan Diaz: 27 at the time of the 1507 expedition