Ian Mortimer has compiled an extremely useful table setting out which fourteenth-century chronicler said what about Edward II's death, and when, in chronological order. For what follows, I have consulted this table and the 2010 biography of Edward by Seymour Phillips; for much more detail and if you're interested in this issue, I strongly recommend reading the books in question (see below for details).  The only chronicler in the south-west of England in September 1327 - he was 90 miles away from Berkeley, in Exeter - was the royal clerk Adam Murimuth, who wrote up his chronicle a few years later. He gives the cause of Edward II's death as suffocation, gives the date as 22 September, apparently a day too late, and claims that the murderers were Sir Thomas Gurney and Sir John Maltravers, when in fact the parliament of November 1330 named the killers as Gurney and William Ockley. Maltravers was never, at any point in his long life (he lived till 1364) accused of any involvement in Edward II's death by Edward's son Edward III. Murimuth and other chroniclers who name Maltravers as one of the men responsible presumably were confused on this point because Maltravers was sentenced to death in November 1330 for his role in the entrapment and judicial murder of Edward II's half-brother the earl of Kent eight months previously. Why Murimuth gives the cause of death as suffocation, what his source was if indeed he had one, or whether he was guessing and thought this was the most plausible method, I don't know, and although he's demonstrably mistaken on a couple of other points, he's usually a pretty reliable source. Elsewhere, Murimuth also says that Edward was "murdered by a trick", per cautelam occisus, whatever he means by that; Professor Seymour Phillips suggests it is metaphorical, meaning that Edward was killed treacherously.
Other chroniclers reasonably close in time to September 1327 give the cause of death as a grief-induced illness (a continuation of the Brut in the early 1330s: enmaladist...grevousement de grant dolour et morust) or possibly strangulation (Lichfield Chronicle, early 1330s), or merely state that Edward died either without further details, say that it was a natural death, or say that he was murdered without stating how (Annales Paulini, Anonimalle, Wigmore chronicle, Newenham annals, Canterbury chronicle, Peterborough chronicle, French chronicle of London, Lanercost, which states that Edward "either died naturally or through the violence of others"). As late as the early 1360s, Sir Thomas Gray of the Scalacronica could still write that Edward "died, in what manner was not known." The Bridlington chronicler of the 1330s wrote that he did not believe "what is now being written" about Edward's death, almost certainly an indication that he had heard the rumours of the red-hot poker (which Thomas Gray decades later either hadn't or thought was too implausible even to mention) but gave them no credence. Note the essential point again that there are plenty of fourteenth-century chroniclers who do not give the red-hot poker as the cause of Edward II's death.
The first mention of the infamous poker appears in the longer continuation of the Brut in about the mid-1330s, which names Thomas Gurney (whose last name is spelt 'Toiourneye') and (wrongly) John Maltravers as the murderers, and includes the detail that a large table was placed on Edward's stomach as he slept and a "spit of copper burning" inserted through a horn inside him and "oftentimes rolled therewith his bowels" (that bit makes me want to vomit). The story is also given in the Polychronicon of Ranulph Higden, a monk of Chester, and in the chronicle of Geoffrey le Baker of Oxfordshire, both written around 1350. It's le Baker, in fact, who gives most of the details of Edward of Caernarfon's supposed mistreatment at Berkeley Castle prior to death; he was constructing a narrative of a saintly, Christ-like Edward nobly suffering the torments of lesser men, with a view to promoting the former king's canonisation. As for Higden, he was summoned before Edward III's council in August 1352 and ordered "to come with all your chronicles and those in your charge to speak and treat with the council concerning matters to be explained to you on our [Edward III's] behalf."  As Ian Mortimer points out, Higden "was more responsible for the spread of the story about the death of Edward II being due to a red-hot poker than anyone else", given that the Polychronicon exists in more than 160 manuscripts and was, like the Brut, frequently used as a source by most later chroniclers. 
After a few decades, chroniclers had no new information to add and of course no real knowledge of what had happened at Berkeley Castle in September 1327, and just copied from earlier chroniclers, whether their accounts were reliable or not. I've already written a post demolishing the often-repeated myth that John Trevisa, chaplain of Thomas, Lord Berkeley, had inside knowledge of Edward's fate in 1327 and that his translation without comment of Higden's Polychronicon with its red-hot poker murder is indirect evidence that the story must be true. (Sigh, if only some modern writers would just do a modicum of research before mindlessly repeating nonsense from others about Trevisa being a child in Berkeley village in 1327 and later being the confessor of Edward of Caernarfon's guardian.) The Geoffrey le Baker story of the red-hot poker murder with all its lurid details is by far the most detailed and probably the one best known today, while Christopher Marlowe's c. 1592 play about Edward II has done more than anything else to popularise this story of Edward's supposed death. As I and others in the comments noted in this post, and as far better historians than me have also pointed out, the red-hot poker story is extremely, highly, overwhelmingly unlikely to be true, and if you see anyone anywhere stating it as certain 'fact', they're only demonstrating their lack of familiarity with the whole issue. As has often been stated in modern times, the red-hot poker story may "have been part of a popular tradition that fitted the punishment to the crime and imagined the ex-king as having been effectively sodomized to death."  My own belief is that if Edward II really did die at Berkeley Castle in September 1327, he is far more likely to have been sedated and then suffocated, which would also leave no marks and was a method of murder the perpetrators knew would work, as opposed to pointlessly sadistic and far-fetched notions of 'let's punish the sodomite by raping him with a hot poker'. Roger Mortimer escaped from the Tower of London in August 1323 by feeding his guards drugs in their wine that knocked them out, so we know he was able to procure sedatives (which, given Edward II's enormous strength, are likely to have been necessary to overpower and kill him).
In the next post (posts?), I'll take a look at the evidence for Edward II's possible survival past 1327: the Fieschi Letter, William Melton's letter, the earl of Kent's plot, William le Galeys and so on.
1) Ian Mortimer, 'The Death of Edward II in Berkeley Castle', English Historical Review, cxx (2005), p. 1209 (reprinted in his Medieval Intrigue: Decoding Royal Conspiracies (2010), pp. 92-93, 107). Wikipedia's current page on Edward II states "it was generally believed, he was murdered by an agent of Isabella and Mortimer on 11 October 1327, although Edward's death is commemorated annually at Berkeley Castle on 21 September."
2) Ian Mortimer, 'Sermons of Sodomy: A Reconsideration of Edward II's Sodomitical Reputation' in The Reign of Edward II: New Perspectives, ed. Gwilym Dodd and Anthony Musson (2006), pp. 58-60, reprinted in in Mortimer's Medieval Intrigue, pp. 55-58; Seymour Phillips, Edward II (2010), pp. 560-565.
3) Antonia Gransden, Historical Writing in England II: c. 1307 to the Early Sixteenth Century (1982), p. 43.
5) W. Mark Ormrod, Edward III (2011), p. 67. See also Ian Mortimer's 'Sermons of Sodomy' article cited in note 2 above and Ormrod's 'The Sexualities of Edward II', also in Dodd and Musson.