The degree that you choose does not really matter. What matters more is how well will you do for your degree, say 1st Class Honours ? That is the step that will play a factor in determining your first job. Subsequently, your career progression and limitations will depend more on your ability, relevant experience and connections rather than your degree. Of course, if you managed to secure a 1st Class Honours, it can help you to get into a management trainee or associate programme and your career progression may be faster.

There are a lot of other companies besides shipyards that are involved in drilling rigs. If you study marine engineering, you are probably limiting yourself to be hired by the shipyards or marine industry only. It's less likely for oil and gas companies to hire a marine engineering graduate. For that versatility, I would choose offshore engineering followed by naval architecture. Shipyards would have no problem in hiring people from the three different degree programmes for their fresh graduates.

Whether you work in the design office or on the rig itself depends on the position that you are applying for rather than your degree. Say if you apply to be a design engineer, you will definitely be more involved in the design. Now contrast this with that of a production or a field engineer.