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Interesting question. I'm no expert. I don't even do much baking, but I was under the impression that the chicken eggs we use today are generally bigger than the chicken eggs from 100 years ago. So you could try buying small eggs, but 16 eggs in a cake does still sound like a whole lot.

I tried quickly researching it to see what I could find. The section on "Egg Sizes [in] the USA" here is kind of interesting, although it says there's no simple answer as most older cookbooks didn't remark on egg sizes. Although there's a quote from an 1896 Fannie Farmer cookbook where she mentions "a pound of eggs (nine)". So that's something concrete you could go on. I haven't tried weighing eggs, but you could give it a try to see how your eggs compare.

They might have had smaller eggs. The USDA has official standards for labeling eggs based on weight per dozen. Those 16 eggs in the cake recipe, I'm guessing, would be wildly out of proportion if you used large eggs, but if you used small or peewee eggs the cake would be on the eggy side though not bizarrely so.

I can't judge the pudding recipe since not all pudding recipes call for eggs, and I have no idea of the yield or the amount of the other ingredients.

Though it does seem strange that a pudding recipe calling for 2 eggs and a cake recipe calling for 16 eggs should be in the same cookbook.

As the cake recipe seems to call for a pound of each of the other ingredients, could it be that the 16 eggs were also supposed to weigh a pound - i.e. a sort of Scottish equivalent of the French Quatre-quarts?

Cetainly they must have been small eggs by today's standards of 2 oz per large egg

I was thinking that too - pound cake is called that because the original recipe called for a pound each of flour, butter, sugar and eggs. But the eggs would have to be very small for 16 of them to weigh only a pound.

Eggs are also sized by weight throughout the UK and Europe, but I think only since about 1950

Eggs were smaller back then, old egg racks are made for what we would now think of as medium eggs,  and it is possible that the small hardy hens kept in Scotland laid particularly small eggs, but they certainly weren't as small as half the size of current eggs, a UK  large egg is now > 60 grams so roughly 8 to the pound, much the same as US sizes  

But it seems that either this recipe writer kept bantams or just liked eggs! The basic cake mixture I was brought up on was to weigh 2, 3, or 4 eggs depending on the size of cake required and then use the same weight of butter, sugar, and flour, so a variant on pound cake

i found the book online and I see it's one of those collections of recipes submitted by individuals, so there's a real possibility of errors

That explains how those two recipes can appear in the same cookbook. It's like a typical community cookbook of today.