In The Greatest Traitor, Ian Mortimer claimed that Edward II's death was faked and in The Perfect King he produces more evidence that this was the case. As well as inconsistencies in chronicles and reports of his death, Mortimer also cites the Fieschi letter which recounts Edward II's movements in Ireland and France after the reported date of his death. Mortimer goes on to report the existence of a William le Galeys in the accounts of the royal wardrobe, who claims to be Edward III's father. This, he claims, points to the deposed Edward II living under the protection of Italian gentry as a recluse. He backs this up with a lot of research concerning Edward III's links to the Italian nobility and the "arrest" of William Galeys to be be presented to Edward III. Mortimer's work is too detailed to recount entirely here, but he accuses the historical establishment of insularity and groupthink when first presented with his findings in 2004 - so has anything changed in the last ten years? Does anyone agree with Mortimer or have his findings been debunked?
I mostly know Edward II for his remarkably unsuccessful military campaigns and to a less extent his problems with nobility. I'm not familiar with Ian Mortimer's argument specifically but it sounds quite far fetched. At a rough glance it sounds like he's failing to be properly critical of his sources. Reports of deposed monarchs living on after they were killed are hardly unheard of. I don't have a reference on me but I'm fairly certain something similar happened with Richard II and in a modern context there is the always famous case of Princess Anastasia.
in 2010 Seymour Philips published the first major academic biography of Edward II as part of Yale University Press's English Monarch Series. I have not had a chance to read it yet (it's in the queue...the endless endless book queue...) but it would almost certainly have a detailed answer to your question. It's a huge volume that took over 20 years to write, very impressive overall.
Bonus: I looked up The Greatest Traitor as I'd never heard of it and I love that the blurb on Amazon contains this gem "He removed Edward II in the first deposition of a monarch in British history." Always good to know that Edward the Confessor and Harold don't count. William the Conqueror should obviously be known as William the Definitely Legitimate Inheritor.
Edit: I just realized that while I don't own a copy of Seymour Philips' book I do own Mark Ormrod's biography of Edward III from the same series (published 2013). I also haven't read it (yet...so many books!) but a quick skim of it of the relevant section does include a reference to these survival myths for Edward II. Ormrod rejects them but I'm only just reading the section now. I will post further information about his argument once I've finished.
Edit2: Ormrod concedes that it is theoretically possible that Edward II survived, but that the evidence is really quite tenuous. The evidence by and large is the letter from Manuel Fieschi, written in c.1336 so nine years after Edward II's death and six years after Edward III had effectively taken control of England, which Edward III may or may not have even ever read. William le Galeys does appear to have existed, and was briefly included in the royal train as far as Antwerp where he left again in 1338, but he would hardly be the first person to claim to be a king he wasn't and there's no evidence that Edward III ever believed him. Ormrod seems to believe that Edward III's policy of forgiveness and reconciliation with the individuals responsible for his father's deposition probably encouraged these kinds of myths, presumably people expected he would want bloody revenge had he truly believed his father to be dead. Ormrod also cites that rumors of Richard II's survival plagued Henry IV for years, in many ways worse than for Edward III because at least Edward was his predecessor's rightful heir. There is also little evidence that these myths were ever particularly popular in England, at least not after Edward III took power in 1330, they seem to be a continental novelty.
Ormrod also provides several references to articles that more directly refute Mortimer's argument which I can copy if you're interested.
I discussed this with an archaeologist/castleologist who has been working on Berkeley Castle (Gloucestershire, UK) for the past 10+ years after I read the first book. Berkeley being where Edward II was kept prisoner prior to his unfortunate meeting with a poker (supposedly).
It was a number of years ago so I cannot remember the details precisely, however he was quite short in saying that it is highly unlikely given the historical record. He cited documents relating to the pursuit and capture of the men that murdered Edward II, to which he gave far more credence than Mortimer's sources.
I accept that this is not the best nor most informative source but it may point you in the right direction to find answers.
(As an aside, does this mean that Mortimer has dropped his assertion from The Greatest Traitor that Edward II was kept at Corfe Castle to be used as leverage against Edward III?)
EDIT: formatting and grammar