At the moment, the room is a pale blue with a strip of dark blue. I want to paint the whole thing cream. I think (and hope!) the pale blue will be OK but I don’t know if the dark blue will show through.
Do I have to put a primer on the whole wall, or just the dark blue section? Do you have any other tips?
The color should not show through if you apply the two finish coats appropriately. Check out Karl's "<a href="https://www.house-painting-info.com/interior-painting.html">Interior Painting</a>" section for excellent pointers.
Now, the texture that is the current edge of the two colors meeting may be something of a concern. Try sanding the line which divides the colors.