How much constitutional power does the UK's Her Majesty's Loyal Opposition have?

Answer 1

They have the power to vote on whether a Parliamentary Motion should become a Bill or not- when a Motion is tabled in the Commons, ALL MPs from all parties have the power to vote on whether or not it should be passed and become a Bill. If the Governing party tables a motion that is controversial within it's own ranks, there is a danger that some of it's members may side with the Opposition and vote against it- in such a scenario, if the Government were to lose the vote, this can lead to a Vote of No Confidence in the Prime Minister and a change of Government leadership.

Answer 2

First of all, the United Kingdom, unlike the United States, does not have a formal, written constitution, so to ask what the constitutional power is of any group in the UK is incorrect. A better phrasing would be simply what traditional powers or what current powers are vested in "X" group.

As for Her Majesty's Loyal Opposition, they are basically powerless. Their main activity is to debate with the Government (the Party or Parties in the governing coalition) to try to force a compromise on legislation and to use the power of their publicity to bring shame to the Government should they enact a policy that the Opposition disagrees with. If the Government is made of a coalition, the Opposition can also attempt to break up the coalition and form a "counter-coalition" where the Opposition controls more seats than the Government. This can create a "Hung Government" because the Government does not have enough votes to pass its legislation and the Government will usually call for snap elections in a bid to re-establish power.