In the US, we say slice of ham as well. It is understood that bread cannot be sliced as thinly. At the deli, we would order ham sliced thinly, in thin slices, or in thick slices. (But not sliced thickly. I don't know why.) A very thin slice may be a shaving (shavings of ham), but a person may expect that the shavings may no longer be the full size of a slice (in area).
Here's a rough guide to size:
slice > shaving > shred > grating
You may also hear of:
chip: chipped beef, chipped ham. like thick shavings, more like wedges.
chop: pork chop, beef chop. like a small steak.