Thats a hard question. It depends on how badly the Aztecs are shaken after killing Cortez. Here are two options. I'll deal with the Portuguese after.
1) If the Aztec leaders come out thinking they did ok after a few setbacks, the Aztecs are going to fail pretty quickly. They'll be more antagonistic towards the Spaniards and will probably have as little as possible to do with the Spanish other than sacrificing any Spaniard stupid enough to get near them.
Expect the Spanish to trade with the surrounding non-Aztec groups. So the enemies of the Aztecs will trade their silver and gold for swords, knives, and armour. They'll trade other things as well, but considering the warlike nature of the area, weapons will be in high demand. The Spanish will probably avoid trading guns except to highly favoured leaders in small numbers.
The Aztecs would find themselves on the losing edge of the arms race, as their enemies attack them with iron weapons. They'd be attacked because they have built up a lot of bad blood in the region, and the attackers will need more gold to buy weapons from the Spanish. So by attacking and conquering the Aztecs they're killing two birds with one stone.
2) If the Aztec leaders decide they came too close to dying for comfort, they may open up limited trade with the Spanish. Using the Merchant class they could set up a trading area, away from their main cities, and heavily guarded. If the Spanish come with apologies and gifts concerning the unfortunate Cortez, they can begin normalized trade with the Aztecs.
In this case the Spanish would probably trade with most of the major groups. Again trading swords and knives, but only sharing guns and gun powder with especially favoured groups.
The Spanish would get rich, and the various groups in the area will each try to get the most weapons and gifts. The Flower Wars in this case, become a lot more violent, and will likely leave the armies of several of the city states devoid of soldiers.
The Spanish may in this case offer their services as mercenaries for concessions and virtual ownership of the city and region.
In both cases disease is going to play a big part in what happens next. Since the Spanish won't be using the Aztec or other people as slaves, except for a few small areas, the diseases will hit hard, but not as hard as OTL.
As the population dies there will probably be a lot of the aristocracy dying. In this case who becomes leader is important. Do they change leaders peacefully or does it go down like the Inca with a civil war. If its peaceful, the Spanish will probably move into a few of the coastal cities and towns. For the majority of cities they will probably just keep trading, and simply move priests in with the promise of helping to end the disease if they become Christian.
If theres a civil war, the Spanish will offer their support to the side that will give them the most. So a de facto takeover. Less messy for the Spanish, and less work for them as well, just as profitable.
Eventually they'll decide to take over like they eventually did in Asia. How well they do depends on how the various city states react to them, and when the Spanish do it. Too hard to tell what happens exactly.
Now the Portuguese may or may not be important in this. If the Aztecs are hostile to all trade, the Portuguese can't do anything, the Aztecs die. In this situation they could support the other city states and groups though. In this case just read the next paragraph about the Aztecs and change the names.
If the Aztecs are shaken and decide to build up through trade, the Portuguese have a chance to get in on the action. Now why the Portuguese would want to support a group against Spain I don't know, but assuming they do, this is a way it might happen.
With the growing arms race in Mesoamerica, each side would be looking for an advantage. The Spanish wouldn't really care as long as they get the gold and any leader who tries to stop it gets removed. So if one group accidently pisses off the Spanish they're in serious trouble. That is a perfect opening for the Portuguese to move in, selling swords, guns, and other things.
Or if there is a civil war with one side supported by Spain, the Portuguese can support the other side.
The problem here is that the Portuguese have to worry about Spain discovering this double dealing. If Spain finds out Portugal has been undercutting them, or selling to their enemies in Spanish 'Territory', it could lead to war. And Portugal will lose.
Something the Portuguese might try is to get some English traders to act as their front men. Not actual Englishmen who support England, just some traders looking for a quick buck. If the Spanish hear about these traders they think England is behind it, or if they catch a trader, well the Englishmen are all liars, of course they'd blame Portugal.
As each side gets more allies and puppets, expect to see a more secretive situation that was seen in India between France and England. Each country controls parts, and they try to expand their influence through trade deals, proxy wars, and assassinations. Spain would have the advantage, but depending on how gung ho they are it may not use it to the fullest extent.
If the situation lasts into the late 1500's, the English will use it to their advantage, openly allying with some of the city states and groups.
Then the Portuguese may begin to openly trade with some of their puppet states.
Expect European wars to break out very soon. If their are some strong rulers in Mesoamerica, they might be able to use the rivalry to remain semi-independant, like some Asian states did in the 1800's. But most of them will become colonies or protectorates of the various European powers.
North America may very well be left alone since all the known gold is in the south. So at first the Atlantic sea coast will see minor towns and ports, primarily related to supplying safe harbours for the treasure ships, raiders, and armies of the European powers.
This may give the Dutch and some of the smaller countries a chance to set up long term colonies in North America, far away from the main action.