Likelihood that Queen Anne of Great Britain has a child that survives into adulthood.

[What if]

No. The succession laws are that time were (and had been since the late Tudor period) male preference primogeniture. Anne was William and Mary's heiress and her surviving children would come after her in the line of succession.

As they say the more you know.

 

[Jürgen]

No. The succession laws are that time were (and had been since the late Tudor period) male preference primogeniture. Anne was William and Mary's heiress and her surviving children would come after her in the line of succession.

From what I understood there was talk about making her husband king, but he pretty much did everything he could to ensure that he would only be prince consort, and it was fully conscious from his side he really didn’t want to be king.

 

[Emperor Constantine]

Well Ungrateful Daughters by Maureen Waller (fantastic book on the family dynamics of the English royal family leading up to the Glorious revolution and into the reigns of the last Stuarts) mentions the health of Queen Anne's eldest daughters (Ladies Mary and Anne Sophia), and the results of their autopsies after their deaths from smallpox. "
The autopsy showed that the elder girl was 'all consumed, but the younger quite healthy, and every appearance for long life'." So while Lady Mary's health was quite fragile and wouldn't have likely lived much longer (even without the smallpox), the doctors seemed to have believed that Anne Sophia had a good shot at a long life. So an Anne Sophia that never gets/recovers from smallpox would most likely succeed her mother as Anne II in 1714.

@Kellan Sullivan is absolutely right about William, Duke of Gloucester though. Looking at his brief life and his major health problems (he is believed to have contracted meningitis at birth, definitely suffered from hydrocephalus, and couldn't walk up stairs unattended at age five, to name a but a few) William was on course to be Britain's bewitched King. Now even without the meningitis, its hard to guess if Gloucester would be healthy or not, but considering the fates of the majority of his siblings (stillborn, miscarried, and died after a few hours), his chances aren't great.

@What if , there's no legal way to bypass Anne in favor of her son. Gloucester's claim came from his mother, therefore she would have to come ahead of him in the succession.

 

[Emperor Constantine]

From what I understood there was talk about making her husband king, but he pretty much did everything he could to ensure that he would only be prince consort, and it was fully conscious from his side he really didn’t want to be king.

King consort, not King regnant. At best George would have a status similar to that of the husbands of María I and María II of Portugal; they were titled King and somewhat shared their wives reign, but their reign's ceased at the Queen's death, and they had no claim to the succession.

 

[Kellan Sullivan]

@What if , there's no legal way to bypass Anne in favor of her son. Gloucester's claim came from his mother, therefore she would have to come ahead of him in the succession.

Actually W3 was looking into cutting Anne out of the succession even BEFORE Gloucester died. He approached JFES with a plan about adoption (twice, both times the offer was rejected) and when determining the Protestantism of the succession, threatened to leave the entire lot (Britain and Holland) to Sophie Charlotte of Hannover, queen of Prussia, if the Electress of Hannover wouldn't give a clear answer on whether she would or would not. He made no mention of cutting Anne out in such an instance, but the way he worded it was to make it seem as if Sophie Charlotte would succeed W3 DIRECTLY