I'm thinking something along the lines of Henry V dose not contract Dysentery during his campaign of which allows him somehow to live a long and productive reign and England wining the Seige of Orleans leaving the French once again very demoralised. A few combat situations later the French sue for peace or are forced to, this ends the 2nd Lancastrian Phase as an English victory. Skip forward a century or two, the northern regions close to the isles are increasingly centralised towards Westminster through a number of reasons such as landownership and trade. Somehow England maintains an island mentality through til Industrialisation despite becoming Kings of France as well.
During this period the Kingdom of Scotland is left vulnerable and over the generations becomes further and further more like a slave of Westminster. At some point due to whatever reasons, a Union of the crowns occurs between England and France with Scotland becoming apart of England and later on become seen as nothing more the a region of England much like one may view Brittany.
Come the Reformation, the still ruling Lancasters for some reason convert to say Calvinism of which spreads like the plague throughout all the held territories. This occurs much faster in the coastal north and so because of such reasons as formation of regions to improve governance, the region is cemented in English hands with the Williamite (otl Dutch) dialects of English becoming the language of the new Calvinist tradition in the region. While the remainder of France converts slower and maintains a strong Catholic tradition to later become the Ireland of the continent.
The United Kingdom of England and France becomes a great colonial power of which dominates the world and by the 20th Century has more then its fair place in the sun. Today Greater English culture is far more conservative linguistically due to the ultra-protestant patriotism created from the Reformation. West Germanic dialects form a sprachbund from Dublin to Berlin.
That is my mildly fantastical view on how such is possible, now I just need Kynan to create a genealogical chart for some 600 years of Lancastrian children.
Just a side note with the Lancastrian continuation being tied with Calais and that. Would their dynasty be referred to as the Lancastrian dynasty today or would they use Plantagenet?
How very nationalistic of you
Hmm if Henry V survives and becomes King of France, his attention is going to be seriously divided between the two, which could lead to alienation amongst his people, and of course depending on how well he deals with French resistance as well, and how he deals with raising his son. But anyway assuming this happens, well, for a union of Scotland and England, one need only ensure that a daughter is born to either James III or James IV, and have that daughter be the heir, or go otl route ala Mary Queen of Scots.
Regarding what they'd be called, I think they'd still see themselves as the House of Plantagenet but historians might refer to them as House of Lancaster.
A rough tree of the mainline of Kings:
Henry V of England II of France reigned 1413-1458
Henry VI of England and III of France reigned 1458-1478
Edward IV of England and I of France reigned 1478-1508
Henry VII of England and IV of France reigned 1508-1538
And so and and so forth
Coincidentally, this also means no War of the Roses, meaning a more stabler England.