1589 was pretty near the peak of support for Arabella's claim: she was introduced at court in 1587, and Elizabeth flirted with naming Arabella her heir between then and 1592. I'd say she'd have a pretty fair shot. She's 14 at this time, so there would be a regency: probably Burleigh as Lord Protector or leader of a Regency Council, and Bess of Hardwick as Arabella's guardian.
The Grey claimant would be Edward Seymour (son of Catherine Grey), styling himself Viscount Beauchamp. He probably has a bit of support, but nowhere near enough to rival the Cecils (Burleigh and his family) if they decide to follow Elizabeth's apparent intent and support Arabella. Seymour's claim is severely weakened by the irregularity of his parents' marriage: they married in illegally and in secret, the only witness died before testifying, nobody could produce the priest until decades later (well after our POD), and the Privy Council had declared the marriage invalid and Seymour a bastard. In Seymour's favor, he's an adult male, and he's already married and produced an heir and a spare.
The Stanley claimant would be Margaret Stanley (nee Clifford), Countess of Derby. In 1579 she was disgraced and banished from court due to unproven treason charges (for trying to have the date of Elizabeth's death divined, and possibly also attempting to poison Elizabeth). Her husband and son still have a fair amount of money and power, but I doubt she'd have much luck if she tried to claim the throne.
James probably would make a play for the English throne, and he might succeed if the Cecils decide to support him, but Arabella's in a much stronger position than IOTL, and James has had much less time to work on building an English support base as unofficial heir presumptive.
It matters a lot exactly when in 1589 Elizabeth were to die. James's OTL marriage to Anne of Denmark was negotiated in August 1589, and the ceremony was held in November. If Elizabeth dies before August, the obvious solution to the succession would be to marry Arabella to James and crown them jointly.