Early instances of 'last name' in the sense of 'surname'
Google Books and Elephind newspaper database searches yield ten matches for "last name" in the sense of "surname" during the period 1782 to 1841. A number of these early instances come from British sources. In both British and U.S. sources, "last name is significantly less common than "surname" during this period. certainly less frequent than instances of "surname." Here, in context, are the ten instances of "last name."
From John Muirhead, Dissertations on the Foederal transactions between God and his Church (Kelso, Scotland, 1782):
The patriarch's [Abraham's] first name was [four Hebrew characters] : His last name [five Hebrew characters] The former imports an high father ; the latter, as explained by God himself, means a father of many nations.
From Charles Wilson, The Wandering islander; or, The History of Mr. Charles North, volume 1 (1792):
Before I quit the house, I must take notice that Thomas Ap Richard Ap Howel Ap Jevan Vychan, Lord of Mostyn, and his brother Piers, founder of the family of Trelacre, were the first that abridged their name, and that on the following occasion : Rowland Lee, Bishop of Litchfield, and president of the Marches of Wales, in the reign of Henry VIII, sat at one of the courts on a Welch cause, and wearied with the quantity of Aps in the jury, directed that the pannel should assume their last name, or that of their residence ; and that Thomas Ap Richard Ap Howel Ap Jevan Vychan should for the future be reduced to the poor disyllable Mostyn, no doubt to the mortification of many an ancient line.
From a review of Henry Bromley, A Catalogue of Engraved British Portraits, from Egbert the Great to the Present Time in The Gentleman's Magazine (London, March 1794):
Presuming, then, upon the propriety of this method, I have observed, as a necessary consequence, in the recital of the titles or designations of the articles of persons of eminence, to state the highest rank which they at any time enjoyed, although such title or rank should not be expressed in the inscription of any of their prints. In like manner, the female sex, whose portraits have been drawn when single persons, when married are described by their last name.
From a theatrical review in The Monthly Mirror (London, June 1798):
June 7.—This theatre [Covent-Garden] closed with He's much to blame and Rosina. Rosina by a young lady, being her first appearance. This young lady's name—that is, her last name (for she has had many) is Francis, and comes from the Royalty Theatre. Our report may be given in five words:—some voice—no taste—a bad eat—abundance of affectation—and no lack of impudence.
On eight other cards, write the same names, with this restriction, that the first name, on each of them, must be taken from that card of the other parcel, whose last name begins with D, the second name from that whose last name begins with 1, ...
From "Reform Dinner at Brighton," in Cobbett's Political Register (London, March 2, 1822):
Thus is the wonder [of Cobbett's influence] explained ; and for this we are indebted to Mr. Scarlett. His name, unhappily for him, has no such charm. His first name, Peter, having five letters, and his last name having eight ; names having nothing "singular" in them, and numbers nothing necromantic! and, accordingly, we find it agreed on all hands, that Peter Scarlett is no conjuror, and has not "such influence," I will not say upon "classes of society," but upon any half dozen men in the kingdom, who have the reputation of having brains in their skulls.
From a notie in the [Vevay, Indiana] Weekly Messenger (August 3, 1833):
Permit me to caution you respecting your tickets. I have seen a number of tickets in circulation, on which my last name is spelled 'Herick,' by whom and for what purpose, those tickets were printed, I know not. See, that my name is spelled BELA HEAH1CK.
From "Fashionable Phraseology," in the New-York Mirror (August 16, 1834):
But further, if the phrase is to be a plural one ["the Misses Browns" as an alternative to "the Miss Browns"], there is, (as I alleged above,) another argument against the the substitution and addition. It is, that the phrase, "the Misses, whose name is Brown," is, in itself, a most gross violation of sense and grammar. If the additional unnecessary words are insisted on, they make another word necessary. The sentence is not English, unless you say, "whose common name," or "whose family name," etc., or something equivalent to that. For each person has one name, and two or more must have names; though the last name of one may be the same as the last name of the other; i. e. though their last name be common to them all.
From Frederick Thomas, East and West: A Novel, volume 2 (Philadelphia, 1836):
"...Did I understand you to say that Ralph was your name, sir—Ralph Beckford?"
"Yes, sir," replied Ralph, "that is my name."
"Well, Mr. Beckford, knowing as I did that your last name was Beckford, I had the impression, my young friend, that your first name was Ralph. ..."
From a translation by the Rev. J.J. Holroyd of Essex of George Lessing's "The Traeasure" (Colchester, 1838):
Raps: Yes, my good old gentleman, I must tell you—now attend to me: if you were to begin quite early, as soon as the morning breaks, with my first name, and go along as quick as you could, I would wager a good sum that the sun would long be down before you could arrive at the first letter of my last name.
From James Fenimore Cooper, The Deerslayer (1841):
"Well, Deerslayer, I'm not one of them," answered Hetty, simply ; "Judith likes soldiers and flary coats, and fine feathers ; but they're all naught to me. She says the officers are great, and gay, and of soft speech; but they make me shudder, for their business is to kill their fellow creatures. I like your calling better; and your last name is a very good one—better than Natty Bumppo."
Two or three of these instances, such as the one in The Monthly Mirror of June 1798, may be using "last name" in the sense of "most recent name," although I would have expected a woman's first name (however recently acquired) to be spelled Frances rather than Francis. But most of these occurrences clearly use "last name" in the sense of "surname."
Ngram results for 'surname' and 'last name'
With regard to the relatively frequency of "surname" in the American English corpus and in the British English corpus, the difference is not great.Here is the American English Ngram chart for the period 1790–2008:
And here is the British English chart for the same period:
As these charts show, for the period from 1840 to 2008, the frequency of occurrence of "surname" in the corpus American English has ranged between 0.00030% and 0.00015% and in 2008 was at 0.00020%. For the same period, the frequency of occurrence of "surname" in the corpus British English ranged between 0.00060% and 0.00025%, and in 2008 was at 0.00025%.
And here are the corresponding charts for "last name" over the same period for the corpus American English:
and for the corpus British English:
In the case of "last name," the frequency in the corpus American English rises from about 0.00004% in 1930 to 0.00024% in 2008, and the frequency in British English rises from less than 0.00002% in 1930 to about 0.00006% in 2008.
These charts invite several inferences:
"Surname" has maintained a fairly steady (though gradually declining) frequency in both British English and American English between 1840 and 2008; and as of 2008 appeared at surprisingly similar frequencies in Google Books' corpus British English (0.00025%) and its corpus American English (0.00020%).
As of 2008, "last name" was roughly four times more frequent in American English (0.00024%) than in British English (0.00006%).
The increase in frequency of "last name" between 1960 and 2008 is approximately 5X in the corpus American English and 3X in the corpus British English.
To sum up, here are the answers I reach to the questions that the poster asks:
Have American English speakers always used the term “last name” instead of “surname”?
No. They have sometimes used "last name" instead of "surname"—but they have also used "surname" quite a lot. Both terms appear to be widely used in American English today. As of 2008, Google Ngram data indicate that the frequency of "last name" (at 0.00024%) is slightly higher in the corpus American English than that of "surname" (at 0.00020%).
It is by no means clear, however, that "last name" appears only in situations where the meaning "surname" is intended. For example a search for "last name on the list" for the years 2004–2008 turns up four matches, in none of which is "last name" used in the sense of "surname." In contrast, "surname" necessarily appears only in contexts where "surname" is intended. So the relative frequency of "last name" and of "surname" in Google Books search results does not constitute a reliable apples-to-apples comparison.
How long it has been this way?
Evidently, for as long as there has been a United States, there have been some American English speakers who said "last name" and some ho said "surname"—and very possibly some who said "last name" on some occasions and "surname" on others.
If AmE speakers ever used the word "surname," when did they switch to using "last name"?
Between 1850 and 2008, the frequency of "surname" in the corpus American English declined by less than 0.0001 percentage point; over that same period, the frequency of "surname" in the corpus British English by almost 0.0002 percentage point. So perhaps a better question would be, "Why did the frequency of usage of "surname" decline at more than double the rate in BrE than in AmE?"
It is true that by Google Books' calculations the frequency of usage of "last name" in the corpus American English has increased almost tenfold since 1900 (and a bit less than threefold in the corpus British English). But given that very little of that increase is mirrored by a decline in the use of "surname," I suspect that the rise of "last name" is largely independent of the fortunes and usage of "surname" in written American English.
In my highly conjectural view, the likeliest source of such an increase is the proliferation of preprinted forms (and references to preprinted forms) that request the name of the person filling them out and that may tend to specify "last name" rather than "surname" because their creators think fewer people in a demographically diverse group will be unsure about what a "last name" is than about what a "surname" is.