Not Prince of Wales

Duke of Cornwall makes sense. But perhaps after Henry Tudor took the throne, the tradition could be changed to Prince of York and Lancaster.

When, in January 1486, Henry VII married Elizabeth of York, daughter of Edward IV, the red and white roses of Lancaster and York merged into the Tudor rose, a symbol of reconciliation and new beginnings.
Likewise for the heir to the throne, the title of Prince of York and Lancaster is a merging of the House of York and the House of Lancaster.
 
Nope. Those are Duchies, Lancaster was, by that point, a title held by the monarch and the Yorkist cause was still too prominent during the early Tudor period to really be adopted.

Henry Tudor married Elizabeth of York to unite the Houses in order to bring peace. My suggestion here was to style the heir to the throne as the heir to these houses. In doing so, the Royals reaffirm their commitment to the union of the houses of York and Lancaster.

If Henry hadn't married Elizabeth of York, then the Yorkists wouldn't have supported Henry's claim for the throne to unseat Richard.

The idea was styling them Princes of the Houses rather than the geographical regions they are named after.
 
Henry Tudor married Elizabeth of York to unite the Houses in order to bring peace. My suggestion here was to style the heir to the throne as the heir to these houses. In doing so, the Royals reaffirm their commitment to the union of the houses of York and Lancaster.

If Henry hadn't married Elizabeth of York, then the Yorkists wouldn't have supported Henry's claim for the throne to unseat Richard.

The idea was styling them Princes of the Houses rather than the geographical regions they are named after.

What's the need? The Tudors are rebranding anyway with a new Royal House, basic genealogy doesn't need a fancy title to affirm that such a union happened, and it just looks insecure to create a completely new title from whole cloth to 'reaffirm their commitment' to a union which has already been affirmed in the most personal way possible (a marriage at the highest level) and which nobody was suggesting breaking apart (well, apart from some minor Yorkists).

Ironically if the title of Prince of Wales didn't exist and the Tudors wanted a more prestigious title than 'Duke of Cornwall' for the heir to the throne, then they'd have probably just created that title seeing as the family from Wales.
 
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