The physics
Light moves more slowly in glass. Put a clear sheet of thickness t and index μ over one slit. It adds extra path to that slit's route.
extra path = (μ − 1)t
The central fringe sits where the two routes are equal. So the whole pattern slides toward the covered slit. The fringe width stays the same.
shift = (μ − 1)t D / d fringes moved = (μ − 1)t / λ
Light that arrives tilted by α gives one slit a longer route, d sin α. The central fringe then moves along the tilt, by D sin α.