Let the PoD be that Elizabeth doesn't send troops north of the border to help the Protestants in 1560, and without her aid, the Protestants are defeated and Scotland remains Catholic. James VI is born six years later as per OTL, but ITTL he's naturally raised Catholic rather than Protestant. Elizabeth still doesn't marry, leaving James as her closest living relative when she dies in 1603.
Could James still become King of England ITTL? On the one hand, the hereditary principle was adhered to quite strictly in this period -- even Mary had made no attempts to pass over Elizabeth in favour of a Catholic successor, for example -- but on the other hand, Protestantism had had a good generation or more to bed in by the time of Elizabeth's death, and the country might not accept the possibility of a Protestant monarch. On the other other hand, a Catholic Scotland would make a good base for Catholic missionaries and efforts to smuggle priests into the country, so Catholicism might be doing better than IOTL, particularly in the north of England.
Could James still become King of England ITTL? On the one hand, the hereditary principle was adhered to quite strictly in this period -- even Mary had made no attempts to pass over Elizabeth in favour of a Catholic successor, for example -- but on the other hand, Protestantism had had a good generation or more to bed in by the time of Elizabeth's death, and the country might not accept the possibility of a Protestant monarch. On the other other hand, a Catholic Scotland would make a good base for Catholic missionaries and efforts to smuggle priests into the country, so Catholicism might be doing better than IOTL, particularly in the north of England.