Supermarkets in Antigua are few and far between and you probably will not be able to find the brands your child is used to in Europe and the USA etc. In Dickenson bay there is quite a large First Choice Supermarket, which sells many brands of nappies, including Pampers and Huggies. You can also find baby wipes here. These are a little more expensive than the UK but they are available if you need them. Swim nappies are not widely available.
Gerber, which is made by Nestle, is available in First Choice and in some other smaller supermarkets. This is also available in the USA.
There are a number of powdered milk brands available including some lactose or soy free ones.
Always use bottled water for preparing bottles and for keeping little ones hydrated. The tap water is chlorinated and generally safe to drink but the difference in how it is treated may cause stomach upsets.
You can buy pasteurized milk in supermarkets which is safe to drink.