I think you have your dates confused. Friesland and East-Frisia were never part of Burgundy, unless you mean West Friesland (which is now part of North-Holland), which was as part of Holland Burgundian. Still easiest way to do it is probably an Anglo-Dutch personal union including Friesland.
Actually, he is half-right and you are half-right. It's a bit of an interesting story. First things first - you are right that Frisia wasn't part of Burgundy. Indeed it was notionally independent and wasn't even a cohesive state. It was instead a collection of towns and villages with little or no central government. The closest they came to a central government was individual towns exerting varying levels of control over the towns and villages around them. The boggy land made the land very defensible meaning they resisted a number of attempts to subdue them.
HOWEVER, they weren't fully independent. They were in fact notionally under the ownership of Holland. The Counts of Holland in (IIRC) the 1200s had been granted the feudal overlord status over Frisia by the HRE in a bit of a "if you can subdue them they're all yours" agreement. Holland attempted a few times to control them but failed, and the agreement didn't entirely stop other powers (such as Oldenburg) from attempting to annex the territory but to all intents and purposes, Frisia was legally Hollandais.
Now, Holland was also independent, BUT the Dukes of Burgundy claimed sphere of influence over it - so Burgundy by proxy claimed overlordship of Frisia. So here's the interesting thing in terms of this whole thing: Holland during the reign of Henry V (of England) was ruled by a woman called Jacqueline of Hainaut. As the name suggests, she also ruled Hainaut (another Dutch territory in what is now roughly the middle of the French-Belgian border) and a third territory people commonly forget, called Bavaria-Straubing. She was unpopular in all of her territories because she was a woman and seen as a weak ruler because of this. In Holland she was forced to fight the Cod Wars against the Burghers who resented her rule. In Bavaria-Straubing at the same time she was forced to fight a war against a rival claimant who eventually took the territory. Because of this, she married Henry V of England's brother, Humphrey Duke of Gloucester, in 1423 hoping that the English connection would give her the troops to win her wars. This greatly angered the Duke of Burgundy, who attacked and forced her to give up all her remaining land, which was virtually the start of Burgundy's accumulation of all the Dutch states which became the Provinces of the Netherlands. Henry had even warned his brother Humphrey not to marry Jacqueline, but he simply waited for Henry to die.
Now, if you modify the HYW around the reign of Henry V you might be able to make this work. You'd need a POD which either makes Burgundy too pro-English to complain, or puts Burgundy literally at war with England so that England can legitimately use the Holland connection to attack Burgundy - AND they must be capable of winning this war. Neither of these PODs is particularly likely but they are vaguely feasible. Maybe a POD involving England siding with the Armagnacs? Then you can have Holland be ruled by Humphrey, and then conquer Frisia. To make it part of the UK you would probably need their thrones to be merged into the same person - perhaps by having Humphrey take the throne of England after his three elder brothers die childless (OTL only Henry V had children of the four male sons of Henry IV so it's not a push) or having the Hollandais comtal family marry into the English throne to fall into Personal Union later down the line. Now, the major thing here is that you get stuck with an England/Frisia which also controls Holland, Hainaut and possibly Bavaria-Straubing (plus anything else they own in France TTL - it can't be "nothing" because they'd never hold the Dutch territories if pushed out of France). Now, this makes for an interesting and eclectic mix of territories to defend, but it also perhaps makes the group of states just too diverse to ever form a single country. You therefore may want England to lose Holland fairly quickly but somehow keep Frisia - perhaps the Frisians swear allegiance to England rather than be overrun by Burgundy, I'm not sure. This is all assuming of course that "Frisia be part of the UK" means "Frisia is part of a single country" and not "Frisia ruled by the same man who rules England" (i.e. Hanover and the UK).
Anyway, talk over. Hopefully some ideas. This is one of the PODs which has personally interested me a fair amount previously, though I must admit I eventually abandoned it as being just too unlikely.