Given that Edward II's father and son knew English, it is highly likely that he did too. Edward was criticised by various fourteenth-century chroniclers for enjoying the company of the lowborn, which is borne out by other evidence: he went on holiday for an entire month in the autumn of 1315 with "a great concourse of common people"; he drank in Newcastle with an unnamed but evidently lowborn woman in 1310; he dined privately in 1325 with a group of carpenters, and a group of sailors on another occasion; he went to a forge to talk to his blacksmith John Cole in 1323; he spent what seems like excessive amounts of time in the 1320s chatting to fishermen and often bought fish from them "with his own hands." There are numerous other such examples. Carpenters, fishermen, blacksmiths and the like would not have spoken French, and I find it hard to imagine that Edward would have taken as much pleasure as he obviously did in the company of the lowborn, and spent as much time with them, if he'd had to rely on interpreters to communicate with them. Although direct evidence is lacking, it seems to me that Edward II must have enjoyed a fluent command of English.
I find it interesting to see that English words occasionally crept into French texts of the early fourteenth century. I've written about Edward's Household Ordinance of 1318, where 'hackneyman' and 'cup house', among others, appear in English. In Edward I's 1306 order that the countess of Buchan should have one or two English women to attend her in her cage on the walls of Berwick Castle, the word 'English' is written in English (Englesche) rather than in French (engleis). Edward II's chamber account of 1325 records a gift of forty-seven caged goldfinches for his niece Eleanor Despenser, written in French with 'goldfinches' in English: xlvij Goldfynches en vne cage. Other English words crept into the chamber journal, such as 'shipbord' and 'shipwreghtes' (shipwrights). I wonder why; was there no equivalent French word, or did the clerk writing the journal just not know it?
To what extent, or whether at all, Edward II and his court used English as a means of communication among themselves is impossible to say for sure. Various scholars claim that although French was the daily language of the law courts and of baronial administration, by about the middle of the thirteenth century it had become an acquired language in England, and that most French speakers were native speakers of English; Michael Prestwich describes the French spoken in England in this period as "an increasingly artificial language, lacking the vitality to change and develop," in contrast to English.  The marvellous Henry of Grosmont, duke of Lancaster, Edward II's kinsman and one of my favourite people of the fourteenth century, wrote a treatise in French called Le Livre de Seyntz Medicines, the Book of Holy Medicines, in 1354. He wrote at the beginning "if the French is not good, I must be excused, because I am English and not much accustomed to French" (si le franceis ne soit pas bon, jeo doie estre escusee, pur ceo qe jeo sui engleis et n’ai pas moelt hauntee le franceis). Obviously this was a literary device to demonstrate Henry's modesty, as his French was completely fluent, even cultured. I can only speculate as to the fluency or otherwise of Henry's English, however, and it's worth pointing out that, whether he spoke much English in his daily life or not and whether it was his first language or not, French, in the middle of the fourteenth century, was still the language he chose to write in. It's interesting to compare Hugh Despenser the Younger, whose many extant letters are all in French, with his grandson Thomas, Lord Berkeley (1353-1417), for whose benefit John Trevisa translated the Polychronicon from Latin - into English, not French.
I'm going to end this rather hotch-potchy post with a couple of paragraphs about a text called Le Tretiz, The Treatise, written by Walter de Bibbesworth in about 1250 for Dionysia (Denise) de Munchesney, the stepmother of Joan, countess of Pembroke (1230s-1307), who was the granddaughter of the great William le Marshal and married Henry III's French half-brother William de Valence. The Tretiz was intended to teach Dionysia and her children the specialised French vocabulary they would need for estate management and husbandry. To quote from the Oxford Dictionary of National Biography, it "consists of approximately 1134 rhyming octosyllables and survives in sixteen manuscripts...The French vocabulary denoting the practices of rural life is equipped in many of the manuscripts with English glosses, the bulk of which stem from the author." What makes the Tretiz so fascinating is the insight it gives into what Bibbesworth expected Dionysia to understand of the French language; he assumes that English is her native language and states that she is already familiar with the fraunceis ki chescun seit dire, "the French which everyone knows" ('everyone' meaning everyone of her class, not everyone in England, presumably), but that he needs to teach her the "French which is not so common," fraunceis noun pas si commoun. The text is in French, and any words which are assumed to be unfamiliar to Dionysia are glossed into English. Bibbesworth himself, a knight of Essex, says that he himself had acquired French in a similar manner. 
Here's an example page from the Tretiz. I love line 256, Louwe oule, chein baie, with the first two words glossed into English as wolfe (wolf) and yollez (howls) and the fourth as berkes (barks). Notice that Bibbesworth didn't feel the need to gloss the third word, chein, as 'dog', evidently assuming that Dionysia would know such an everyday word. In lines 250, 275 and 282, every French word is glossed into English, but other lines appear only in French with no English translations: a good indication of the level of French Bibbesworth expected Dionysia to know. As for Edward II a few decades later, he seems to have been entirely comfortable speaking French in public and to have been fluent, articulate and confident in the language - and for sure he must have frequently spoken it in private with the French Isabella and the Gascon Piers Gaveston - but was French his native language, or did he learn English as his mother tongue from the people around him? Was he completely bilingual? I wish I knew...
1) Michael Prestwich, Edward I, p. 6; Marc Morris, A Great and Terrible King: Edward I and the Forging of Britain, p. 8; N. Denholm-Young, Richard of Cornwall, pp. 86-87; T.W.E. Roche, The King of Almayne, pp. 134-135.
2) Ian Mortimer, The Perfect King: The Life of Edward III, Father of the English Nation, pp. 199-201, 265.
3) Seymour Phillips, 'The Place of the Reign of Edward II', in The Reign of Edward II: New Perspectives, ed. Gwilym Dodd and Anthony Musson, p. 225, note 30.
4) C.M. Woolgar, The Senses in Late Medieval England, p. 101; Michael Prestwich, Plantagenet England 1225-1360, p. 557.