It’s a stretch to say Elizabeth was ‘hostile’ towards them.

Mary of Scots had a claim to the English throne via her grandmother Margaret Tudor, sister of Henry VIII. She was Catholic and considered legitimate, whereas Elizabeth was Protestant and considered a bastard in Catholic eyes. Mary of Scots was raised in France where Henry II declared and raised her as an heir apparent to the English throne. After she married the dauphin, Francis, her coat of arms displayed her declaration to be Queen of England, France, and Scotland and Henry II declared them the King and Queen of England. Which was a slap in the face to Elizabeth I, who had just succeeded Mary I to the throne. Once Francis died and Mary of Scots was forced back to Scotland, Mary toned down her antagonism of Elizabeth and urged instead that Elizabeth should make Mary her heir. Elizabeth wasn’t inclined to name an heir who might then be used in a conspiracy against her, and not Mary at that moment, who had already tried to declare herself queen over Elizabeth.

Mary and Elizabeth grew a warmer relationship and sweetly wrote letters to each other and nearly met, and Mary sought advice from Elizabeth about her desire to marry. Elizabeth for various reasons suggested Robert Dudley, which Mary took as an insult. Meanwhile Henry Stuart, Lord Darnley, had made his way to the Scottish court where Mary took a great interest in him and wed him shortly after. Darnley had a claim to the Scottish throne, and as he shared Margaret Tudor as a grandmother with Mary of Scots, a claim to the English throne. It was against custom for a member of court to marry without the monarch’s permission, and nearly treason for someone with close royal blood. So for Darnley to marry behind Elizabeth’s back, and for Mary of Scots to take Darnley as a husband, was a big deal to the English court. It cemented William Cecil’s belief that Mary was perpetually trying to dethrone his queen. Even so, there was a point where Mary wrote to Elizabeth and said that if something should happen to her, she wished Elizabeth to raise James, which Elizabeth found touching and convenient.

Then Darnley was involved in a plot to overthrow Mary, then Mary was accused of helping to murder Darnley, then Mary married a man involved in the plot to murder Darnley while letting off suspected murderers. Her popularity took a huge drop, and the instability and poor decisions did not make her a good candidate for heir. The fact that she was Catholic and England was a Protestant country with Protestant nobles was also not in her favor. Mary eventually made her way to England to seek Elizabeth’s protection, and that was a liability to Elizabeth. She debated on how to proceed while Cecil and other members of her council urged her not to assist Mary back to the throne. If Mary was indeed guilty of Darnley’s murder, she couldn’t be put back on the throne, but Elizabeth didn’t want to be forced into helping and in a mock trial she found Mary neither guilty nor not guilty of Darnley’s murder. After some conspiracies against Elizabeth in Mary’s name, the Babington Plot happened and it was seen as Mary agreeing to have Elizabeth murdered so Mary could claim the throne. In the meantime, James VI had grown up rather ignoring Mary, having been raised to believe she was a murderess and adulteress. There came a point where Mary asked James to co-rule with her, and Elizabeth instead offered to recognize James as sole ruler and hinted at considering him her heir. It was James that chose Elizabeth over Mary. Another claimant to the throne was Arbella Stuart, another relation of Margaret Tudor, and she was whispered to be a possible heir to the throne. Although it’s likely it was never very serious. Elizabeth didn’t know from day one of her ascension that she’d never have an heir, and it wasn’t as if she didn’t believe there wouldn’t be one. But she was someone who did not like to make decisions until the very last moment, and she didn’t want to name an heir who could be used against her. She also liked to have the heir slot as a card to play, because she could keep James in line by not confirming if he was heir, or acting as if she might give it to someone like Arbella instead. Plus, depending on who she named as heir, it might give the impression early on that she would never marry, which wasn’t what she presented in a portion of her reign.

Mary of Scots drew first blood when it came to the relationship between the two queens. Sometimes they called each other sister, and Elizabeth didn’t like the idea of a queen being dethroned by Parliament, let alone executed. The affair set a bad precendent, and she very much believed in the right of dynastic rule. Mary shot herself in the foot as far as ever being a really viable heir for Elizabeth. And, it was James who set his own mother aside to choose Elizabeth as a patron and keep the Scottish throne for himself. Even while imprisoned by Elizabeth, Mary was still the annointed Queen of Scotland and James shouldn’t have been able to bypass her when she’d never ‘willingly’ abdicated. Elizabeth was reluctant to declare an heir, but she wasn’t out to get her Scottish cousins or fight against them. She reacted to circumstances and did her best to protect her rule.