There's no direct connection
Note: This post is spoiler heavy, and using spoiler tags would make it rather cumbersome. If you don't wish to find spoilers, this probably isn't the answer for you.
Ragnar from "The Last Kingdom" is Ragnar Ravnson. Ravn being his father. Ravn is the blind man Uhtred meets and cuts food for shortly after being captured. Ragnar and his father Ravn (along with Ragnar's sons, Ragnar Ragnarson and Rorik Ragnarson, and his daughter Thyra Ragnarsdottir) are fictional characters who are not claimed to be in any way related to Ragnar Lothbrok.
The similar names are a coincidence because of common names (or more likely, not a coincidence: Ragnar was a fairly common name in Denmark in the 5th-10th century), and the creator of the Saxon Stories likely took the name from the stories of the "real" Ragnar Lothbrok.
- Ragnar Ravnson is Ragnar, son of Ravn
- Ragnar Lothbrok is Ragnar, and Lothbrock is actually a nickname
In both cases, we'd expect their sons to be called Ragnarson/Ragnarsson, as Nordic naming convention at the time used that style.
So no, they aren't related.
Ragnar Lothbrok's family features prominently in "The Last Kingdom"... just not the ones you were thinking of.
Ubba Lothbrokson. The guy Uhtred kills beside the sea in Wales, is Ragnar Lothbrok's son. He and his brothers Ivar and Halfdan led the Great Heathen Army which invaded Britian in the time of Alfred the Great. We'll see more of Ivar in the next series of the show.
Note that in the Saxon Stories the brothers are referred to as either Ubba/Ivar/Halfdan, or occasionally "Ubba Lothbrokson" etc. Typically we'd expect them to be called "Ubba Ragnarsson" etc, rather than "Lothbrokson", but I assume this would have been confusing considering we had another prominent Ragnar in the same stories, and with Uhtred calling himself Uhtred Ragnarson, along with Ragnar Ragnarson. It appears that the author of the books decided, to avoid confusion, to go with the "Lothbrokson" convention, rather than the more historically accurate but potentially confusing "Ragnarson" convention.
Note that in reality we aren't actually certain whether Ubba, Ivar and Halfdan were brothers, nor whether they were Ragnar Lothbrok's (sometimes written Lodbrok) sons. However in The Last Kingdom, Bernard Cornwell (the author of the Saxan Stories) decided to run with the idea that they are.
The sons of Ragnar Lothbrok
- Ivar The Boneless: The fictional Ragnar, who captured Uhtred, served Ivar. We'll see more of him next season, I assume. Ivar is the most likely of the three "brothers" we see in the Uhtred stories (Ubba/Ivar/Halfdan) to actually be real and also be related to Ragnar Lothbrok, along with Sigurd and Bjorn who are not seen in the Last Kingdom series. If Ragnar existed in the way that is believed, Ivar was his son. He was certainly the son of someone called Ragnar, but we aren't sure how accurate the legends of his father are.
- Ubba: Uhtred just killed him. Definitely existed, and led the Great Heathen Army with Ivar and Halfdan but we aren't sure if he's related to Ragnar Lothbrok and Ivar.
- Halfdan: I believe Halfdan is in Ireland at this stage of the Last Kingdom series (at least, he is in the books). He invaded Northumbria with Ubba and Ivar in the attack that took York (Jorvik/Eoferwic, the capital of Northumbria). Uhtred's father died shortly after this while attempting to re-take York. As with Ubba, he definitely existed but we aren't sure if he's related to Ragnar or Ivar
- Bjorn Ironside: Never believed to have invaded or spent much time in England, or at least if he did he's ignored by Bernard Cornwell in the Uhtred stories. Spent time raiding France and Italy. Along with Sigurd, almost certainly did exist and, if Ragnar existed, was probably related to him.
- Sigurd Snake-in-the-Eye: Attacked Northumbria along with some of his brothers, but then we aren't sure exactly what happened to him. Doesn't feature much in the Uhtred stories, but is referred to. We're fairly sure he existed, although he's the only of the sons with a question mark there.
In short, we're fairly confident all 5 of the "sons" existed, and we're fairly sure that a "Ragnar" existed, who was Ivar's father.
Things get a bit blurry after that. Was Ragnar the man we see in these stories, or was he "just" Ivar's father, and a more typical man? Ivar is almost certainly related to him, with Sigurd and Bjorn being less certain but quite likely. Ubba and Halfdan are bigger question marks - they may be related to Ivar, or something may have gotten lost in translation over whether they were blood related or more of a "brothers in arms" type of relationship.
Either way, as the father of 1, 3, or 5 of the most influential viking Jarl's, Ragnar's likely to have had some... embellishments... made to his character and achievements.
Interesting side-note: It wasn't Ragnar who pulled the "pretend to be dead, ask to be buried in a Christian graveyard" trick, it was his (probable) son Bjorn Ironside, during his raids in Italy: he used it to gain entry to the town of Lucca.
So why don't things quite match up?
There's a clear overlap between the two stories, but why are some characters missing? Well partly because both series embellish their stories, change history a little etc - simply because there is not much written history (from the Danish side, at least) at this time. So while there's a lot of overlap, they both ignore some of the legends while acknowledging other parts. This leads to some inconsistencies (such as Sigurd, who in Vikings will raid Northumbria, while in the Saxon Stories will practically be ignored). The lack of Danish written history makes it hard to know anything with much certainty: there's a surprisingly comprehensive written history on the "English" (which didn't exist yet)/Saxon side, but by it's very nature that history is susceptible to being second hand, hearsay/rumour or mistranslated/misunderstood.
We don't even know for sure that Ragnar Lothbrok existed (Ragnar Ravnson, on the other hand, definitely being fictional). He probably did, but much of what we know are unconfirmed stories. Legend has is that Ubba, Ivar and Halfdan (who all definitely did exist and led the Great Heathen Army) were his sons, but we aren't sure. We also aren't sure if they were all his sons, his only sons or whether he had others. We believe that Bjorn Ironside (who definitely existed) was his son, and there's suggestion that at least some of Ubba/Ivar/Halfdan were too.
So what happened is that Vikings made up some of the life of Ragnar Lothbrok (Who may or may not have existed), and The Last Kingdom made up some of the life of Ubba/Ivar/Halfdan, while assuming that they were indeed related to Ragnar and were brothers of Bjorn Ironside etc. Between the two, they put together a story that's grounded in history, but probably only half right.
In short, we've got two partially-fictional series about some people we don't actually know much about. The stories overlap, although we aren't totally sure if the characters they're based on overlapped in the same way.