BP shares fell in London and New York amid fears that the company's la

A is the right answer.

Meaning: BP shares fell in London and New York amid fears that the company's latest efforts to contain the spill may falter.

B is incorrect because of the wrongful placement of that in the sentence after New York erroneously suggesting that the that-clause modifies New York or at best London and New York and this is illogical at best.
C, D, and E can be eliminated based on the unjustified use of the past perfect tense had fallen. The use of the past perfect tense erroneously suggests that the sentence presents a sequence of events in which the fall of BP shares is an earlier event. This is not the case in the sentence.

BP shares fell in London and New York amid fears that the company's latest efforts to contain the spill, which started three months ago with the sinking of a leased rig, may falter.

A. BP shares fell in London and New York amid fears that the company's latest efforts to contain the spill, which started
B. BP shares fell in London and New York that amid fears in the company's latest efforts to contain the spill, which started
C. BP shares had fallen in London and New York amid fears that the company's latest efforts to contain the spill, which started
D. BP shares had fallen in London and New York amid fears that the company's latest efforts to contain the spill, which had started
E. BP shares had fallen in London and New York that amid fears in the company's latest efforts to contain the spill, which had started