For my answer, I will be heavily referencing the book Hellfire: The Jerry Lee Lewis Story by Nick Tosches. As a warning, this answer is long and requires a lot of context.
For the context of the time period, Elvis had just joined the army and was out of Lewis's way in terms of competition. Lewis himself had just starred in the upcoming film High School Confidential, which would earn $1.4 million in profit ($14 million in today's money) and had just released a single of the same name that just barely missed the top 20. Sun Records was also about to release his first LP. And on top of it all, Jerry Lee Lewis was about to tour England thanks to Oscar "The Baron" Davis; thirty shows, $26,000 ($267,024.50 in today's money), and a whole new market to sell records.
For the context of his marriage, Jerry Lee had married Myra Gale Brown - the daughter of his cousin J.W., making her his first cousin once removed - twice; the first marriage (taking place in December of 1957) was unlawful, as Jerry Lee had still been married to Jane Mitchum, his second wife. The scandal broke out before Jerry and Myra had married the second time in June of 1958. Nonetheless, there had been some concerns among Jerry's circle regarding the wedding; during a pre-tour stop in New York to wait for Oscar Davis, Jerry was approached by Sam Phillips (the owner of Sun Records) and Sam's brother Judd. Both had previously advised Jerry to keep the marriage a secret, but Judd now asked how Jerry was going to approach the press about the matter.
"She's my wife," Jerry replied, "there ain't nothin' wrong about that."
And technically, Jerry was right. First-cousin marriage was not illegal in the state of Mississippi (where the marriage occurred) until 2010, and to this day first-cousin-once-removed marriage is still legal under current Mississippi marriage law. Nonetheless, Judd wasn't buying it, telling Jerry that:
"Right an' wrong don't have anything to do with it, those people ain't gonna like it."
Jerry was adamant, though, and insisted that:
"People want me, they're gonna take me no matter what."
Jerry Lee Lewis was very wrong in this regard. As they got off the plane in London, reporters asked who the young woman was. Despite Oscar attempting to distract them, Jerry revealed that Myra was not only his wife, but fifteen, and that they had been married for two months. The only truth here was that they were married; she was actually thirteen, and it had been five months.
Myra then was asked by the press if she didn't think that fifteen was too young to marry. She replied that:
"Age doesn't matter back at home. You can marry at ten if you can find a husband."
This was also true; Mississippi had no official minimum age, but since the passing of Section 93 of the Mississippi Code 1972 the state had required either parental consent if the girl was as young as 15 or - under certain conditions - a judge's approval if the girl or boy is younger; boys were prohibited from marrying before the age of 17 unless a judge allowed. In this case, Jerry - then 22 - didn't need anyone's permission but Myra's; the night of, Jerry had told J.W. that he was going to take her to see his new movie Jamboree. Instead, he asked her to marry him, she said yes, and they were wed the same night.
Jerry was also fully prepared to marry her, having gotten a marriage license with the help of a female friend in Fayette the day before; there, he stated Myra was 20.
(On top of this, during their remarriage in 1958 in Fayette, Louisiana, Jerry wrote "None" for the question on relation of bride to groom. Louisiana did not set a legal marriage age until 2019, and Louisiana too allows first-cousin-once-removed marriage.)
After keeping the marriage secret for a week, Jerry finally told J.W. and J.W.'s wife Lois. The two were livid to the point that J.W. threatened to skin Jerry alive and annul the marriage. Somehow, though, this did not happen.
Now, how did Jerry Lee Lewis handle the reactions from the British press and listening audience?
By accounts, he seemed more cocky than surprised. A reporter in England asked if Jerry thought the scandal would hurt his career, to which Jerry simply said:
"Back in America, I got two lovely homes, three Cadillacs, and a farm. What else could anyone want?"
And when he got back to the states, he assured American reporters that the English simply did react to rock and roll like Americans did, and said that he "got homesick."
He finally raised his voice when another reporter asked if he didn't think it was a bit odd for a man to marry a thirteen-year-old.
"You can put this down. She's a woman."
And when Myra was asked?
"I think what Jerry thinks about it."
And by the time he and Myra had their first child - a boy named Steve - in 1959, they had bought another home. Tosches had this to say:
"Jerry Lee would sit by the pool in the early evening, at that hour when the sun began to fall and the darkling, sighing colors came. He would sit there and not smile and not think. It was only a matter of time, he knew, till he would once again be redeemed. He would sit there, gazing at the tame, purling water, and he would know that. And he would make a fist, wishing that there were a horse that he could knock to its fucking knees."
First of all, this is not really a mid 20th century historical episode: it's still a thing in the United States, which has one of the worst records and laws in the Global North. 44 of 50 states permit (as of 2021) children to wed if their parents consent. In many states, you can even legally marry below the age of sexual consent, and people do. A [2022 study](https://www.jahonline.org/article/S1054-139X(21)00552-8/fulltext) found, in addition to the 44/50 states figure, that:
"Child marriages violated statutory rape laws in 14 states. The proportion of child marriages that violated statutory rape laws varied from 1% to over 50%. In 33 states, some or all statutory rape laws exempted sex between married couples from the definition of crimes. In these states, the proportion of child marriages that would have been crimes, without these exemptions, varied from less than 1% to over 80%."
Interestingly, in 1950 age restrictions for marriage were much less permissive than today. Every state except North Carolina required a minimum age of 18 (many had 21, specially for males) without parental consent. NC lowered the minimum age for girls to 14 (male still had to be 18). However, there were almost no meaningful age verification requirements, and people lied about their age, so that census figures show much higher rates of underage marriage than marriage records, suggesting that the actual rates of underage marriage were much higher than the declared rates.
A more focused analysis of census data for the period 1935-1969, found that 3.5% of women across the US had married before the age of 16 and 4.5% at age 16 and 7.4% at the age of 17. So chances are that not only Jerry Lee Lewis, but most Americans, would have known people who contracted underage marriages, with 15.4% of women being under 18 at marriage, and 3.5 out of 10 being under 16. 1956 marks the year with the lowest median age at marriage in 20th century USA.
There is in fact a good history of child marriage in the United States by Nicholas Syrett. For the period you are asking about, this is Syrett's conclusion:
"Following World War II, the age of first marriage dipped to an all-century low. Numbers of teen brides and grooms soared through the early 1960s, and then quickly dissipated. While early marriage fit right into a United States bent on fighting the Cold War with domestic stability at home, experts, journalists, and academics also bemoaned the large numbers of high school students who married in these decades... the uptick in early marriage, especially among white urban and suburban dwellers, was caused by conflicting messages about sex, which resulted in premarital pregnancies and shotgun weddings; a nationwide emphasis on domesticity; and by cravings by teenagers for adulthood, symbolized through marriage. While rates of early marriage for rural and nonwhite residents remained steady, the real change here was a white middle-class early marriage surge, which is what resulted in all the expert hand-wringing."
Which is to say that perhaps Jerry Lee Lewis' surprise is not all that surprising. In general culture, this was the peak of child marriage in practice, even as in elite culture it was gaining in (justified) stigma. Nevertheless, marrying a 13 year old would, under normal circumstances, have raised eyebrows even in general society, without necessarily eliciting outright condemnation.
But Jerry Lee Lewis was also already married, so this was bringing a child bride into a bigamist relationship. Although people make much of the consanguineus second cousin dimensions, I don't believe marrying a cousin once removed would have been massively scandalous, and Presidents and famous Americans married closer than that. Bans on cousins once removed are both, recent, and unusual (14 states in America). There are almost no such bans anywhere else in the world.
Rather, the true reason for the scandal, I think, and its vehemence, was the puritanical and confused attitudes toward sex in contemporary America, and the generational conflict signified by Rock'n'Roll.
One major reason why child marriage was (and remains) so weirdly acceptable in the United States, is because the socially conservative fear of unmarried sex and the ritual purity of women and girls is typically dramatically greater in political discourse than any actual concern for the psychological well-being of women and girls. When child marriage was in the service of conservative values, it was considered a decent response to unmarried sex, child pregnancy, or unbridled male lust.
But Jerry Lee Lewis was not inducting 13yo Myra Lewis Williams into acceptable patriarchal society. He was co-opting her into the moral cesspool of Rock'n'Roll. We are used to thinking of sex, drugs and Rock'n'Roll, but this was in fact the first major Rock'n'Roll scandal, the milestone that cemented that association as a trope in cultural history. Jerry Lee Lewis' marriage was a convenient lightning rod for all the generational outrage against the changes Rock'n'Roll was an emblem and symptom of.
Had Jerry Lee Lewis been an insurance salesman, an ad-man, a professional, there might have been some tut-tutting, and that would be that. But Lewis was an icon of moral retreat to his critics, so his child marriage was not been inducted into the sanctity of marriage, but corrupted into the cesspool of generational decline. This was a chance to gain traction in culture war, and hit out not just at him, but at the purveyors of a new, mass youth culture.
In all this, the actual child involved was really peripheral. The driver was moral indignation, not child protection. A good sense of the Zeitgeist may be found here: https://lavocedinewyork.com/en/entertainment/2022/10/31/jerry-lee-lewis-child-bride-and-cousin-the-first-big-scandal-of-rocknroll/
Given the absence of precedent, it makes sense that the virulence and strength of the attack may have surprised Jerry Lee Lewis, in a way that would have been entirely expected and predictable to a Mick Jagger coming into pop culture with the cultural conflicts fully defined.
However, even such moral outrage was largely performative, and had no real implications or effect. Beneath the surface, no one cared. On the contrary, you can see that in the United States, the scandal was exploitable and could be kept going to the benefit of all concerned. Except of course Myra. Conservative personalities and church leaders could grow their constituencies and Rock'n'Roll marketers could grow theirs off this outrage theatre.
US society as a whole was simply not outraged by this. In 1959, just after the whole Jerry Lee Lewis scandal had peaked, Elvis Presley was dating a 14 year old, with her parents' consent, and multiple more 14 year olds without anyone raising a finger.
It was not in fact until they went to London in 1958, that he was actually cancelled. Here cultural attitudes were different. This did evidently come as a shock to Lewis. In London, where the minimum age of marriage across England had been 16 since 1929, the police interviewed Myra to check on her wellbeing. The outrage was not primarily driven by patriarchal notions of purity and culture wars, but by societal consensus that marrying a 13 year old was just creepy and harmful.
You can see the incomparably different attitudes in the two countries from this interview with Jerry and Myra on their return: https://youtu.be/P_0_nXbmAhM
The interviewer has no whiff of outrage, he comfortably addresses Myra as Mrs. Lewis. In contrast, see the quoted Brltish press in the article, or this front page article in the (UK) Daily Mirror: https://www.topfoto.co.uk/asset/7244/
Curiously, it was only after the furore in England that Lewis faced meaningful consequences back home. You can tell from the interview above that even the press upon his return had no sense that he would become a pariah shortly. Somehow, it was not his marrying a child that sunk his career for a decade or more. It was not the fact he had not divorced at the time he married her (as he hadn't when he married his previous wife). It was certainly not that she was his first cousin once removed. Somehow, it was the fact that he scandalised Britain, which tipped the scales back home.
You can understand his being confused by the mixed messages American Society was giving him.
As a final note, in emphasising how English society rejected vehemently his marriage to a child, it is worth noting that it had no massive issues, just a few years later, with the multiple underage groupies associated with major celebrities.