The key is actual the men who dominated in the 1590s - so to be honest it is going to be whether Cecil shifts to James from Arbella as the preferred choice - to the Protestant men of Elizabeth's court there were a number of considerations a) the natural Godly order - that a man should govern not a woman - their affection/fear of Elizabeth not withstanding most of her courtiers accepted that a woman's role was not to rule over men b) In James VI there was a natural male successor who was Protestant, already a reigning monarch, and by 1594 a father of a male heir c) James VI was the senior heir general of Henry VII and had the strongest claim morally.
Elizabeth herself simply did not like most of her relations on her father's side - for obvious reasons they were all at some point rival claimants to her own rights to the throne - she really didn't like her Grey cousins at all and certainly had a long-standing aversion to Arbella and James' grandmother Margaret Douglas. As she lay near to death she really just allowed her senior courtiers to make their own decision by not objecting to James VI - her own consistent refusal to name an heir is unlikely to change. If Arbella marries Seymour (which arguably would unite the rival claims from Mary and Margaret Tudor - strengthening their joint hand against James' superior claim) then Elizabeth is likely to be as furious as when Catherine Grey allegedly married Seymour in the 1560s - so you might see the Queen in anger immediately bring the daughter's of Lord Derby to court (they were strictly the legal claimants under Henry VIII's will).
The reason James VI succeeded with minimal problems was because he had the strongest claim morally - he was an easy heir if you like as all the other's were beset by problems.
For Cecil to switch you would need a stronger alliance between him and the family of Bess of Hardwick I think and in otl he had better family connections with the Derby family.
Succession under primogeniture -
Heirs of Margaret Tudor Queen of Scots -
James VI of Scotland, Henry Duke of Rothesay, Prince Charles of Scotland, Princess Elizabeth of Scotland, Lady Arbella Stuart.
Heirs of Mary Tudor Queen of France Duchess of Suffolk
(declared illegitimate by order of Elizabeth I - Edward Seymour and his issue William Seymour, Francis Seymour, Honora Seymour, Anne Seymour, Mary Seymour)
Lady Anne Stanley, Lady Frances Stanley, Lady Elizabeth Stanley, William Stanley, 6th Earl of Derby and his issue
Succession under the will of Henry VIII
Margaret Countess of Derby (d 1596) Ferdinando 5th Earl of Derby (d1594) Lady Anne Stanley, Lady Frances Stanley, Lady Elizabeth Stanley, William Stanley, 6th Earl of Derby