What is the history of Hnefatafl in relation to Chess and/or Go? Which was developed first and what if anything influenced the development of Hnefatafl?
History of these ancient boardgames is a bit murky due to their really old age.
The wikipedia article on Hnefatafl is a good starting point, but you can already see from that how vague the information is. Hnefatafl being played since around 400 is certainly older than Chess. In fact, it can be considered a predecessor of Chess.
A not so easy relation is that between Hnefatafl and Go - if it exists at all.
What can be said for certain is that written documents on these three games can placed in a timeline like this:
What is particularly interesting is that Chess is believed to have originated in India, from where it basically traveled to Europe. By around 1000 Chess was far-spread in Europe.
As for the initial development of Hnefatafl, I am not aware of the influences. What is known is that Hnefatafl is tied very closely to the Vikings. Due to their enterprising nature they traveled through most of Europe helping to spread Hnefatafl. Also note that the end of the Viking age coincides with the time of the introduction of Chess, so it is not surprising that the latter continued to dominate the European scene for centuries to come.
Edit: In terms of influence between Hnefatafl and Chess, I'd wager neither significantly influenced the other. From the mere timeline it is possible that Hnefatafl may have influenced the development of Chess. But that would be a risky theory, as I am not aware of any written connection between these two and the spacial separation between Vikings and Indians makes it ever so unlikely.
Others have already mentioned the general history of the three games, but I can add a bit about hnefatafl itself. The game seems to have similarities to an earlier Roman game, ludus latrunculorum, which was descended from an ancient Greek game, petteia. Both of these older games shared hnefatafl's straight-line move and its capture method of surrounding enemy pieces on opposite sides.
The Roman game made its way as far as Denmark, and this is where the A.D. 400 date that everybody quotes comes from. It's actually the date of a Roman ludus latrunculorum board dug up in Vimose, Denmark, in 1848, rather than a hnefatafl board (the back of the board bore the squares for ludus duodecim scriptorum, a Roman predecessor to backgammon). Hnefatafl was developed, probably from ludus latrunculorum, some time between A.D. 400 and A.D. 750 which is when the first clear hnefatafl evidence shows up (a set of pieces in Salmo, Estonia).
From this I'd say that hnefatafl doesn't have any direct relationship to chess. Petteia may have made its way to the borders of India with Alexander the Great, and thus seeded chess as well as hnefatafl, but there's no evidence for this. Hnefatafl was chess's predecessor only in its social position, not through any line of descent - chess was a usurper.
As for go, I can't imagine that there's any connection. It wasn't known in Europe till after hnefatafl had passed out of general knowledge. There is a coincidence of the standard size of the modern go board, at 19x19 intersections, with ONE old form of hnefatafl (the alea evangelii, c. A.D. 924-939), and that board from Vimose had 18 squares along one edge, but probably fewer along the other axis.
I do not know about the others, but I can tell a little about the history of Go, one of the oldest board games in existence. There are various sources on Go history, but especially the early stages are widely different in each. The source used here is generally well accepted among Go players.
It is certain that Go was invented in China (called weiqi there). According to legend, an ancient Chinese Emperor invented the game to enlighten his son, though it is unsure if it was Emperor Yao (2357-2255 BC) or Emperor Shun (2255-2205 BC). Other sources (Encyclopedia Britannica) state 2356 BC.
Some even claim Go was developed as a game in China perhaps 4000 years before present time, but this date has been questioned by many. Either way, Go, a game that resembles a war and involves terms such as "territory" and "killing" has always been a popular game to be played at the court, slowly spreading to the people from there.
The oldest surviving reference is found in "Analects" by Confucius from the 6th century BC. Starting in the first century AD, there are texts specifically devoted to Go, namely "Yi Zhi" (Essence of Go) by Ban Gu. The earliest recorded games ("Wu Diagrams") stem from around 250 AD.
Starting 1050 AD, various books (for instance Qijing Shisanpian, the 13-Chapter Classic of Go), some still well known today, were created. Among them were problem collections as well as early works on joseki (patterns that are used frequently), some of those patterns are still in use by professional players today.
Go was imported at an unknown point in time (sometime before the early 7th century AD) to Japan, and gained popularity there as well as in China. About 600 years ago, Go became a widespread phenomenon with Go houses (elite study circles) becoming more and more important, later also gaining increased social and political relevance.
To end this boring story and return to the question: Hnefatafl, from what I gather, is purely Western and was in no way influenced by Go. I also never encountered either Hnefatafl nor Western chess in any texts on early Go history.
However, it is worth noting that there is a chess variant very popular in China. It is called Xianqi and probably originated from the same ancestor, the old Indian Chaturanga, as "Western Chess". It is actually said to be the most popular game in the world with half a billion players. However, it sprung into existence too late to interact with Go.
I see you ask about the origins of Hnefatafl, which are obviously hard to trace since the Vikings had effectively no written history. It is possible that Chaturanga, a precursor of Chess, had some influence; it was certainly played in Persia and Iraq at the time when Viking traders reached there. There really isn't much likeness, though, particularly since it is believed that Chaturanga queens moved like Chess kings.
More likely is Nine Mens Morris, a game with visible similarities to tafl, and which was well known throughout the Roman Empire. (An early version, Three Mens Morris, was known in Ancient Egypt.) I would guess that the long winter evenings in Scandinavia also had something to do with it.