The legal heir to the English throne according to various theories of succession:
Will of Henry VIII
- Anne Stanley is the most likely heir here, as discussed above. May face opposition, since the Stanley family has been under suspicion for Catholic sympathies.
- Edward Seymour, Viscount Beauchamp, was generally regarded as a bastard due to the irregularity and lack of documentation of his parents' alleged marriage. But if he can get this reversed or overlooked, he's genealogically senior to Anne.
Male-Preference Primogeniture
- Arabella Stuart, as discussed above.
Illegitimate Descendants of Henry VIII
- George Carey, 2nd Baron Hunsdon -- his father was widely rumored to be the illegitimate son of Henry VIII by Mary Boleyn. This is an extremely weak claim, but there is recent precedent for royal inheritance through bastard lines: Henry VII's claim to the throne came through an acknowledged bastard of John of Gaunt, and Henry VIII at one point had been setting up his acknowledged bastard Henry FitzRoy as his heir.
York Heirs
- The senior descendant of George, Duke of Clarence (middle brother of Edward IV and Richard III) is George Hastings, 4th Earl of Huntingdon. His older brother (Henry Hastings, 3rd Earl of Huntingdon) had at one point (in the 1560s, I think) tried to set himself up as Elizabeth's heir if she died from illness, and has attracted a bit of support before Elizabeth recovered and quashed the discussions.
- There are some surviving descendants of John de la Pole, the appointed heir of Richard III (descended from Richard's sister Elizabeth, Duchess of Suffolk). I think the senior heir of this line in 1603 is Jacques de Secondat de Montesquieu, a complete non-entity in English politics, as the family has been French for 2-3 generations by this point.
Lancaster Heirs
- Isabella Clara Eugenia, daughter of King Philip II of Spain, is descended by legitimate line from John of Gaunt on both her father's side and her mother's side. The post-invasion plan for the Spanish Armada had been to put her on the English throne.
I think we can rule out Isabella, Jacque, and George Carey. Anne Stanley's probably the front-runner if the Catholicism rumors don't stick, and Arabella Stuart or George Hastings otherwise. Arabella might try to shore up her claim by marrying one of the other claimants or their heir (IOTL, she eventually marries William Seymour, son of Edward Seymour).
There's also an intriguing possibility of abolishing the monarchy after Elizabeth's death. IOTL, there was some discussion among members of Elizabeth's privy council who opposed James as heir (most notably Sir Walter Raleigh) of establishing a Republic upon Elizabeth's death.
As for the Scottish throne:
- James Hamilton, 3rd Earl of Arran had OTL been considered heir presumptive of James VI until the birth of James VI's first son. By 1603, however, James Hamilton has long been confined to his rooms as a lunatic, so he may be passed over in favor of his son John Hamilton.
- Arabella Stuart has a pretty strong genealogical claim based on her descent from the Earls of Lennox, but has little or no native Scottish political support. Her only chance of claiming the Scottish throne would be with English armies at her back.
- Arabella's cousin Ludovic Stewart, Duke of Lennox inherited Arabella's father's Scottish titles, and could be argued to have inherited the Lennox claim to the Scottish throne as well.