John Brown became the Queen's servant in 1864 when the Queen, born in 1817 would have been 47.
I doubt any personal or physical relationship would have been instant so she would have been 50 or more when she had this mythical baby.
Remember this was in the late 1860s/70s. Even today when people, even royalty are healthier and the medical knowledge and care around giving birth has improved exponentially, conceiving a baby naturally over the age of 50 is almost unknown.
I think this rumour must fail under the heavy weight of Queen Victoria simply being too old to conceive a child. when John Brown was her servant.Anyway DNA evidence should be able to prove the truth of the claim.
It is sad isn't it that even today, there is still the assumption that if a man and woman are close friends the relationship must be sexual. Why could John Brown and the Queen not have a very close but platonic relationship.
The King, when Prince Charles, had a valet he was very close to, whome he protected even when proven to have been less than honest at times. No one has ever suggested that this intense personal relationship was homosexual, so why should Queen Victoria be subject to this attack from grubby minded individuals/