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We have done exactley the same trip . Leicester to Southhampton no problem with National Express , they drop you off and then you need a taxi to get you into the docks.
You might be able to get a national express coach, otherwise cunard can usually arrange for coach travel or a private hire car either directly through them or through your travel agent.
I need to get from warrington in cheshire to southampton to catch the queen victoria on the 6th May 2009.