As I understand it, what made the Normanization of English elite culture possible was the pretty much wholesale replacement of the old Anglo-Saxon political elite with a new (Anglo)-Norman political elite by way of confiscation and dispossession of Anglo-Saxon lords after they either fought against William I during the initial conquest or rebelled against him later on, which created an almost uniform control over government and higher church offices by Norman nobles.
Henry VII in contrast regarded reconciliation with the Yorkist nobility as the way to secure his throne and without changing that I cannot see why Henry would decide to promote Welsh culture and language, style himself as culturally Welsh or speak Welsh as it doesn’t gain him anything (he enjoys strong Welsh support regardless) and alienates the very nobility he wants to bind to his dynasty.
However, there might be a way for Henry to change this policy: both Elizabeth of York and Cecily of York die before his IOTL 1486 marriage to Elizabeth of York, one of them after Bosworth.
Anne of York is next in line in order to fulfill his 1483 pledge of Rennes to marry a daughter of Edward IV and unite the Lancastrian and Yorkist claims, however Anne is 11 years old in 1486 and much too young for marriage and is in for a long engagement. Rumors arise that Tudor/Lancastrian supporters killed off Elizabeth/Cecily (whichever of them died after Bosworth) so that Henry can disavow his engagement to Anne at an opportune time and avoid fulfilling his pledge and that he plans to reign in a much more vindictive manner towards Yorkist supporters than previously stated.
As a result, the Simnel Rebellion in 1487 enjoys more widespread support among the English and Irish pro-Yorkist nobility, lasts longer and is a more devastating conflict, in which Henry is forced to rely much more heavily than IOTL on Welsh support and troops, but Henry wins in 1488/89.
After the latest bout of civil war Henry decides that his reconciliation policy has failed and replaces the English lords that rebelled against him with his most loyal supporters, many of them Welsh. The Irish lords that supported the rebellion are mostly pardoned because Henry doesn’t have the means to bring them to heel at the moment.
Henry also voids the agreement to marry Anne of York as it doesn’t gain him much after the policy change (perhaps bribing the Church to forbid the marriage due to frowned upon ‘affinity’ as a smokescreen to get rid off his no longer wanted engagement) and instead looks for a foreign match, perhaps France in order to cement French support for his dynasty and a generous dowry (throwing Brittany under the bus earlier than IOTL) or Spain.
Perkin Warbeck then appears in the Burgundian court and gains Burgundian and later Scottish support as IOTL but the situation is different as disaffection with Henry’s perceived anti-Yorkist and pro-Welsh policies lies a more fertile ground for support in England, especially in the North (even Thomas Stanley as he didn’t gain as much from the aftermath of Simnel Rebellion as IOTL and instead has to share the bounty with new Welsh lords and has been losing ground at court), and so Warbeck lands with Burgundian and Scottish support in Northern England sometime between 1492-1495 and gets some popular support, possibly including local uprisings in other parts of England.
The Warbeck Rebellion is defeated once again with a substantial Welsh contribution and Welsh lords are awarded lands in England (alongside English Tudor loyalists), but the danger isn’t over for the Tudor dynasty as the Irish lords with possible pro-Yorkist sympathies still pose a threat and will need to be brought to heel in the future plus a de la Pole has escaped England and has found shelter in the Burgundian court.
Recognizing that heavy Welsh contribution to his army and the loyalty of his Welsh lords have saved his throne two times, that many of his lords at court are now culturally Welsh, that he needs to make them feel included and to maintain the extraordinary strong Welsh loyalty has enjoyed Henry begins to style himself as culturally partially Welsh and begins to converse in Welsh with his Welsh lords.
Slowly, Welsh language and culture begins to seep into English court life and a new Anglo-Welsh court and elite culture begins to emerge.
