Hi Andy,
Now the Deer you are thinking of are Muntjac Deer, they weigh about 20lbs and stand about 18" high. Now shotguns are only legal for certaind reasons to kill Deer. What that program shows sounds very dodgy to me. If there is a lot of crop damage a land holder can use a shotgun to control Deer but you have to use certain size shot and I believe a permit from DEFRA (Department for Environment Food and Rural Affairs) is required.
Muntjac Doe
Muntjac Doe on a Ride early morning
Pair of Muntjacs, Buck and Doe both yearlings.
My Mounted Mutjac, this is an old boy who was begining to go back, though a respectable size he is not trophy class.
Myself holding a nice Doe shot one evening.
As you can see thay are small, size does not matter in our law they are still Deer and so covered by "The Deer Act"
Hope that helps.