Firstly, there were no lamentations of slaves in the Mississippi delta in the 1930s; the American Civil War in the 1860s ended slavery. Of course, rural black people in the Jim Crow south in the 1930s - including in the Mississippi Delta - weren't exactly free either; they still suffered under some fairly repressive segregration laws, lynchings occurred, and ultimately they were still being exploited on plantations owned by white people (e.g., the Dockery cotton plantation where the blues musician Charley Patton lived).

As to the blues, there's a popular idea that it goes back to the West African roots of the African-Americans transported to America to become slaves, but more likely the blues style was a popular turn-of-the-century style amongst the itinerant musicians who were jobbing entertainers in the region, and which was usually one of many styles of music that the musicians could play. By the 1920s, the idea of the 'blues' had been spread through the U.S., thanks to W.C. Handy's 1914 song 'St Louis Blues' in particular, and plenty of recordings of 'urban' blues singers like Mamie Smith (who recorded 'Crazy Blues' in 1920) and Bessie Smith (which is more obviously in a jazz realm, with prominent horns, and being less focused on acoustic guitar). It's actually not entirely clear that the rural/country blues style is the older one. I mean, there was nothing stopping the rural blues singers like Blind Lemon Jefferson from having heard recordings of the urban blues and having it influence their music.

The original swell of interest in country acoustic blues - what you're probably thinking of - occurred in the late 1920s, with the likes of Skip James, Blind Lemon Jefferson, and Son House being recorded as part of a trend by record companies hoping to record ethnic folk music to sell record players, basically, to ethnic people in different parts of the U.S (including in the rural black south). To their surprise, there turned out to be a big market for Blind Lemon Jefferson in particular, and so record companies hoped to cash in, recording a whole bunch of different rural black singers with acoustic guitars shortly before the Great Depression kicked in and they (almost) all went bankrupt.

In particular, blues recordings by these musicians were popular amongst record companies (and consumers) during this period, because it was relatively easy for singers to make up new material on the spot, rather than pilfer the relatively limited larders of the usual material those musicians might have played to actual crowd (which would probably be a more varied selection, including jazz tunes, and Broadway songs - anything the crowd wanted to hear enough) - thus providing the novelty the popular music market typically demands.

After this brief period of popularity for this kind of music, there were what we would call blues musicians sporadically recorded by record companies like Vocalion (who recorded Robert Johnson in 1936), and song collectors like Alan Lomax (who recorded a young Muddy Waters playing acoustic blues on Stovall Plantation in Mississippi in 1942).

At this stage, there's effectively two ways in which this kind of music makes its way to white audiences.

The first is in the development of a market of (white, urban) collectors of old folk 78rpm records, perhaps epitomised by Harry Smith, who put together the Anthology Of American Folk Music albums in the mid-fifties, which featured a mix of 1920s/1930s recordings of rural folk musicians (both black and white). Amanda Petrusich profiles some of these collectors and their history in her excellent Do Not Sell At Any Price. The idealistic folk revival shepherded in the 1950s by the likes of Pete Seeger and others provided a sympathetic (albeit somewhat ideological) ear for such music, and a justification for listening to that rather than, you know, Frank Sinatra. (A much cleaned up version of) folk music began to gain real popularity in the late 1950s, with folk trios like the Kingston Trio having genuine chart success singing old folk songs they potentially might have learned from the Anthology of American Folk Music and other such compilations. All of this meant that by about 1960 or so, there was a) a market for the Newport Folk Festival (and other such festivals), and b) the people behind these festivals were keen to have old country blues singers on their lineups. Such festivals located several still-very-much-alive country blues singers (Mississippi John Hurt, Son House, Skip James, etc), and put them on the big stage for their music to be listened to by a white audience who were probably there largely to see the likes of Joan Baez and Bob Dylan.

The second route by which the Delta blues got to rock’n’roll is epitomised in the story of McKinley 'Muddy Waters' Morganfield, who I mentioned recorded playing on a Mississippi plantation in 1942. Increasing automation in farming techniques meant that the traditional cotton picking occupation of the black rural South was increasingly outmoded as the 20th century progressed, while (word had it) factories in the North of the US, in places like Chicago, were crying out for more workers, and they didn't care if they were black workers from the South. As a result of this and other factors (you know, oppression and lynchings in the South, for example), there was a 'Great Migration' of many of the rural black South to places like Detroit and Chicago (see the Jimmy Reed blues, 'Bright Lights, Big City').

Muddy Waters himself moved to Chicago in 1943, hoping to become a professional musician, and for a while working in factories to pay his way. Being a professional singer/guitarist in Chicago - a very electrified city - required the electrification of Waters' acoustic guitar and then the purchase of a proper electric guitar with a electromagnetic pickup. For musicians like Waters, there turned out to be a market for the country blues gone electric, with big city backing instrumentation (drums and bass, sax, etc) - after all, electric versions of country blues songs (as Waters' initial run of singles were) quite suited people who had recently moved to Chicago from the rural South, who heard in the music both their past, their present and (they assumed) their future. As such, a genre now called 'Chicago blues' developed, focused around the record label Chess Records, which put out records by Muddy Waters, Bo Diddley, Howlin' Wolf and others. While using electric instrumentation, this music had something of the attitude of the country blues, and was appealingly rough around the edges in comparison to the slick, jazzy productions of B.B. King (of that era) which were aimed at the pop R&B market.

One of the things that the Chicago blues style felt like was kind of like the older, tougher brother of rock'n'roll. This was in some ways not a coincidence - Chuck Berry also was signed to Chess Records. Jackie Brenston & His Delta Cats - famous for what's sometimes considered the first rock'n'roll song, 'Rocket 88' - were called that because Brenston and his band (Ike Turner's Kings Of Rhythm under another name, because Brenston sang the song) were an R&B band from Clarksdale, Mississippi - not from the plantations nearby like a few of the bluesmen previously mentioned, but from the city itself. Attempting to emulate the slick jump and jive bands of the era, they ended up with an upbeat, energetic, but rougher sounding music that would sound to later audiences quite a lot like rock'n'roll.

The success of the Chicago blues sound simultaneously with the somewhat-puritan folk revival led to an odd situation where some of the Chicago blues musicians - Waters, for example - would play one weekend with a loud blues band with bass and drums and loud, distorted guitar, and the next weekend might play a folk festival as an acoustic performer playing traditional folk blues (while often playing exactly the same songs).

Across the mid-to-late 1950s, there was a big fad style of dance music aimed at white teens, called 'rock'n'roll', which often incorporated blues-derived chord progressions and modes of performance. By 1959 or so, the initial wave of popularity for this music was basically over, and fans of the music style either moved on, or moved deeper. For, say, the London, England-based members of the Rolling Stones, playing rock'n'roll was difficult to make happen, as the jazz clubs turned their noses up at it (and it seemed a dead style) - however, the jazz clubs felt that the blues was justifiable, as it was sort of part of the history of game; Keith Richard first got talking to Mick Jagger at a train station because Jagger was prominently holding a copy of The Best Of Muddy Waters (which Jagger had ordered by mail order from Chess Records in Chicago) and Richard had heard of Waters but never actually heard his music.

The enormous success of the Beatles led, fairly quickly, to a British blues scene, as there quickly developed a market for music that like the Beatles, but was perceived as more masculine (as opposed to the Beatles, who of course had teenage girls screaming at them). Thus you get groups like Them, the Yardbirds, the Animals and of course the Rolling Stones making a living essentially doing covers of Chicago blues songs and related styles (with something more of a rock'n'roll flavour).

One of the American blues musicians who went over to Europe in the early 1960s - Muddy Waters or Sonny Boy Williamson II, maybe? - famously went over one year with an electric band, and played to basically horrified audiences who expected a rustic singer with an acoustic guitar. The next year, invited back, the blues musician prepared an acoustic folk blues set. In between, the Beatles and the Rolling Stones had become very popular, and the blues musician played the acoustic folk blues set to disappointed fans who were hoping for some genuine electric Chicago blues.

In a very historical context, the blues is a wildly misunderstood topic. Our modern day understanding of what "blues music" is really only goes back to roughly around the 1950s and 60s, during the height of electric, "Chicago blues", and the blues and folk revival. At this point, what constituted a "blues song" was more defined by it's sound and musical structure, rather than the feeling it is trying to convey. This wasn't really the case in the 1920s and 30s, however. In fact, it was the exact opposite. A blues song was not a song with a 12-bar format, an AAB rhyme scheme, ladened with flattened thirds and fifths. To a 1920s or 1930s, so called "blues musician" or "blues guitarist", a blues song was simply a sad or sorrowful one. The 12 bar structure and the blue notes just happened to be the prevailing and popular song format among early 20th century African American musicians, and was not tied to any particular topic.

For an example of this, take what Sam Chatmon says in this Alan Lomax interview in the late 1970s. He begins talking about one of the very first songs he learned on guitar, "Make Me A Pallet On The Floor", a song that has many stereotypical "blues" elements from a purely musical point of view. Yet he makes a distinction, stating that it is a "blues type", but not a blues. He goes on to describe the first "actual" blues song that he heard, reciting the lyrics only in prose, with them being, "I'm going down to the river to jump and drown, I think about the woman I love and I turn around". These are very particularly depressing lyrics, which contrasts with the lyrics of "Make Me A Pallet On The Floor", which are not really sad or depressing at all, and in fact describe a somewhat "ideal" (But definitely not moral) situation for the speaker (With him sleeping with someone else's wife, and nothing bad coming of this, seemingly).

Keeping this in mind while looking at the recordings of many rural African American musicians from Chatmon's time period, it becomes a bit clearer that not everything they recorded was meant to be a blues, despite most of it sharing a similar musical structure. I doubt, for an example, that Blind Willie McTell would consider, "Your Southern Can Is Mine", to be a blues, nor would I imagine that many of Blind Willie Johnson's much more gospel-esque and hymn/spiritual influenced records were considered to be blues records despite his enthusiastic use of things like slide guitar.

This becomes even more apparent in the discog of certain musicians, such as Blind Blake, Henry Thomas, and various jug band's (Like Cannon's Jug Stompers), who performed things that teetered on clearly vaudeville-influenced stuff, to what borderlined on country hoe-downs, with a lot of definitely "blues" (As in sad) stuff thrown in the mix. In Blind Blake's case in particular, although he likely never deviated too far from his ragtime roots, there's a clear difference between the nature of say, "Diddie Wa Diddie", and, "Early Morning Blues", even down to the titling of the records. One would likely have been considered to be the equivalent of a pop song (Diddie Wa Diddie) or "hokum", the other a blues song (Early Morning Blues).

Now, why this distinction matters in the context of your question revolves around the idea that the blues, in some way, "transitioned from the lamentations of slaves". While certain elements, like the flatted thirds and fifths or blue notes, and the syncopation of some performances, can most likely be traced back to the music of the enslaved Africans and their descendants in the rural south, 20th century African American music (Which is the best way to describe what we tend to think of as blues music) is not slave or African music. It was a particularly African American (And I use "African American" in a deliberately distinct manner from "slaves") music that was a popular, often-times party oriented music that people liked to dance to in their free time.

It's contextual relation to slave music (Or at least what we know of slave music) is very, very superficial. It's relation to anything related to civil rights is just as superficial, if not completely non-existent. The lyrics to this music described very clearly mundane topics anyone of any background could have related to (Once you got past some of the unique slang, that is), or just topics somewhat unique to the individual singer. Understanding this is crucial when discussing the transition of "blues" into rock n roll, which is often seen as a particularly "white" form of this music, distinguished largely by it's topics related to having fun, partying, and (later) doing drugs and having sex, as opposed to the more "down and out" and "oppressed" topics of the old "blues masters".

In reality, this unfortunately common outlook couldn't be further from the case. As mentioned above, not everything performed and recorded by so-called blues musicians in the 1920s and 30s were blues/sorrowful songs. Many were light-hearted songs alluding to thinly-veiled sexual topics, songs that blatantly encouraged the listener to dance to the underlying, funky groove, songs that promoted and mentioned popular dances, and even songs about drugs. In many ways, the music that African Americans were making at that time was already functionally "rock n roll". Once we relinquish this idea that "blues" music was a particularly reflective music that was in anyway obviously rooted in the the very racially tense lives of early 20th century, southern African Americans, it becomes a lot easier to see how young, white college age kids (Such as Elvis Presley, or the members of Canned Heat) in the 1950s and 60s tuned into black radio stations as well as dug up these old records and not only found enjoyment in the music, but found some aspects of the lyrics relatable.

Edit: Spaced the paragraphs and made some minor corrections.