Queen Victoria was not pregnant during Christmas 1846. Her fifth child, Princess Helena, was born May 1846 and her sixth child, Princess Louise, was born March 1848. Her Majesty the Queen never suffered a miscarriage or stillborn, either.
While Queen Victoria did eventually meet Princess Sara Forbes Bonetta, and became her godmother, she did not attempt to raise her in the palace.
By the time the events of the episode happened Ernest was already married having taken a wife in 1842.
When Queen Victoria is running outside in the snow looking for Sarah, snow is falling but there is no snow on her clothes at all.