Furthermore, your link is to the US product.
Poor effort.
If you search via macdonalds.co.uk you will find the GB page. It is not clear if they mean GB or UK, no doubt due to American incomprehension.
https://www.mcdonalds.com/gb/en-gb/product/chocolate-milkshake-medium.html#collapseTwo
It shows the following ingredients for a chocolate milkshake. It does not show American corn syrup. But it does show milk.
Milkshake Base
EITHER: Allergen Ingredient: Skimmed
MILK, Sugar, Cream (Allergen Ingredient:
MILK), Whey Powder (Allergen Ingredient:
MILK), Glucose Syrup, Stabilisers (Guar Gum, Carrageenan, Carob Gum).
OR: Allergen Ingredient: Reconstituted Skimmed
MILK, Sugar, Cream (Allergen Ingredient:
MILK), Allergen Ingredient: Skimmed
MILK Powder, Glucose Syrup, Whey Powder (Allergen Ingredient:
MILK), Stabilisers (Guar Gum, Carrageenan, Locust Bean Gum), Natural Flavouring.
OR: Allergen Ingredient:
MILK (2.9% fat), Sugar, Whey Powder (Allergen Ingredient:
MILK), Glucose Syrup, Stabilisers (Carrageenan, Locust Bean Gum, Guar Gum).
OR: Allergen Ingredient: Whole
MILK, Allergen Ingredient: Skimmed
MILK, Sugar, Whey Powder (Allergen Ingredient:
MILK), Glucose Syrup, Stabilisers (Guar Gum, Carageenan, Locust Bean Gum).
Chocolate Flavour Milkshake Syrup
Water, Glucose Syrup, Alkalised Fat Reduced Cocoa Powder (7%), Colour (Plain Caramel), Fructose Syrup, Caramel (Sugar, Glucose-Fructose Syrup, Water), Salt, Cocoa Mass (0.7%), Butter (Allergen Ingredient:
MILK), Natural Flavourings, Acidity Regulators (Phosphoric Acid, Citric Acid), Emulsifier (Lecithins), Preservative (Potassium Sorbate).
Allergy Advice: For allergens including cereals containing gluten see ingredients in
BOLD