You would probably need to make significant changes in the last decade of Elizabeth's life and to James VI's life to create the right circumstances for a row over the succession.
The majority of her council were quite content with a Stuart succession - James VI was the senior heir general of Henry VII, he was Protestant, a reigning monarch, married to a Protestant Princess and had children.
Elizabeth's own vanity dictated that she felt she should be succeeded by someone of equal rank rather than the rag tag of common born cousins from a junior line.
The only real alternatives were - 1) Arbella Stuart (James' English born cousin and a teenaged girl) her claim was secondary to James and only really held up if James was barred because of his foreign birth. 2) Edward Seymour - son of Lady Catherine Grey - declared illegitimate and it would take an awful lot for Elizabeth to reverse that (he has at least three sons by 1590), 3) Margaret Clifford Countess of Derby (granddaughter of Mary Tudor) d 1596 - her heir was her granddaughter - Lady Anne Stanley (Anne's father Ferdinando died before his mother) (ANNE was the heir in law to the English throne in 1603 according to the Third Act of Succession of Henry VIII)
In terms of foreign claims - most of the key players (the Spanish and Austrian Hapsburgs) had a Plantagenet claim through their descent from John of Gaunts daughters - the senior claim rested with the senior heir general of Phillipa of Lancaster Queen of Portugal - Teodósio II, Duke of Braganza.