Both are actually possible and are relatively interchangeable in this context. However, B is the better answer because it emphasizes that "living in England" was something still in progress at the time being talked about.

B

In addition to the other answers: 'he had lived in England for two months' - is also still correct and would work well in the context of something historical/biographical. This could be describing the life of someone in the past, which is moving on to talk about them leaving England, for example. "At that time, he had lived in England for two months and had decided it was time to move somewhere sunnier!"

B is the correct answer

The second one B, because using "at that time" means that he had lived there more than 2 months