Instead of setting sail from Andalusia, Columbus, or someone else, sets sail from a port in Galica. As the voyage is several longitudes north, the ships instead land in Florida, rather than the Caribbean. As a result, inland explorations begin much sooner, particularly around the southern Appalachian mountains and the Mississippi delta. Early contacts with Native Americans are brutal.
A couple of decades later, contact is made with the Mexica and they are promptly conquered. Meanwhile, British, Dutch and French settlement begins along the eastern seaboard.
Later on, as the Spanish Empire faces economic crisis and military overstretching (as it rolls through South America), the North American territories declare their independence, shortly thereafter, the political centre of gravity of Spanish North America shifts away from the old colonial areas in the east (around OTL north Florida and Alabama), to the more densely populated, and more native/mestizo Valley of Mexico. The new United States of Mexico abolishes the Spanish caste system and introduces a bunch of economic reforms that encourage European investment. The northern, non-Spanish territories are not the lands of oppurtunity they were in OTL, as they were denied the potential to expand westward due to Spanish control of the Appalachians, (the exception being New England, which was free to expand northward). Europeans seeking to escape poverty and persecution begin to settle around the Gulf of Mexico, for the first few generations, there are many divisions in the USM, natives in the west, and Europeans in the west, patriots in the centre and Spanish loyalists in the Appalachians, unassimilated foreigners in the east, and under-Hispanicised natives in the south - to help integrate their society, the Mexicans begin a campaign of north-western expansion, communities from all of Mexico's cultures are granted titles to the unclaimed land in the Wild West.
Around this time, the a new generation of politicians takes charge, those who modelled themselves on the revolutionary founding fathers were replaced by an elite of assimilated immigrants, (Spanish, by the way, remains the lingua franca, while German and Nahuatl are distant seconds and thirds). The new generation are secular, liberal and idealistic, drawing from many philosophical ideas from Europe.
But this generation of politicians were occassionally too radical for their people, (too secular for traditional capitalists, too modernist for the paganistic southerners, too abolitionist for the slave-owners in the north-east, and too intefering for the libertarians in the Wild West), and so, the USM descends into a chaotic civil war in the early 1800s.
After nine years, Mexico City has been taken and retaken several times, and eventually abandoned as a centre of power, the victorious faction, (uber-capitalist, pro-European Westerners), have moved the capital to a port-city overlooking the Gulf of California. The major casualty of this war was the permenant loss of the slave-holding north-east, (which managed to secure its independence due to French assistance - the French are major utilisers of slave labour in their North American territory), however, the rest of the huge country is soon united under a common leadership, (with the exception of the far south, which is divided into ethnic homesteads, and given limited self determination as subject territories). As the 19th century ends, the USM reaches its territorial zenith as a treaty is signed dividing the far north between the USM, British North America and the Russo-Japanese North American Company.
TTL's 20th century is as chaotic as OTL's, there are three major global wars, that Mexico manages to remain neutral in, untill the very end, when they quickly pounce in on the winning side. The three world wars amount to economic suicide on behalf of the Old World powers as their financial and imperial clout collapses. By TTL's 1970s, the USM is the world's sole hyperpower, later in that decade it begins testing nuclear weapons, and uses the implicit threat of ever using them to secure the obedience of the international community. Spanish is the world's lingua franca and the USM leads the world in almost all aspects.
Map key:
Green: United States of Mexico, and subject states
Gold: Republic of Appalachia
Blue: Republic of New France
Red: Commonwealth of New England
Orange: Republic of New Netherlands
Purple: State of Alaska
Teal: United States of New Austria