Henry VII was born in Wales in 1457, and lived there until 1470. As a child he was educated by Oxford graduates while a ward of William Herbert, a Welsh supporter of Edward IV. He would have learnt English and Latin, but I'm unclear on what place the Welsh language might have had in his formal education. He would have encountered the language outside the classroom, but someone else will have to answer whether he spoke it himself.

Henry then lived in exile in Brittany (and eventually France) from 1471 to 1485, from the age of 14. By the 1480s he was reportedly "more comfortable speaking French than English" so if he spoke English with an accent as an adult, it would more likely be French than Welsh.

When Henry returned to to challenge Richard III in 1485 he counted on Welsh loyalty to the Tudor family to bolster his forces and to allow for the spreading of propaganda utilizing Arthurian myths to depict him as the rightful king. So yes, he leant on his Welsh roots to get support, but any attachment to the region seems to be based on expectations of familial loyalty more than a personal sense of nationality. Once he was King emphasis was placed on him being chosen by God and a successor of King Arthur, rather than on his Welsh roots which would have made him appear more of an outsider to his subjects.

I can't say specifically if he was considered Welsh by his English subjects over the course of his reign (hopefully someone else can provide more details), but I hope I've answered some of your questions!

Source: Tudor: the Family Story, Leanda de Lisle